Abstract Algebra Primer

Note: The first year of undergraduate abstract algebra essentially falls into two main categories: group theory and ring/field theory.  Our Abstract 1 course is an introduction group theory and Abstract 2 is an introduction to ring/field theory.  However some schools reverse that order.  From what I can tell from the sample GRE’s I have most of the abstract algebra questions focus on groups, with some basic ring questions.

Basics of Group Theory

A group is a set G together with a binary operation on G (a way of combining two elements in G to get another element in G, sometimes referred to as a “closure” property and written as multiplication) for which:

The operation is associative: ∀a,b,c∈G, a(b c)=(a b) c.

There is an identity for the operation: ∃e∈G, ∀x∈G, e x=x e=x

Every element has an inverse with respect to the operation: ∀x∈G, ∃y∈G, x y=y x=e

Although not part of the definition, the identity turns out to be unique and the inverse of an element turns out to be unique (under multiplication, the inverse of x is written (x^(-1))).  Note that the operation can be written as addition, in which case identity is often written as 0 and the inverse of x is usually written as -x.

There is no requirement that the operation be commutative (i.e. ∀x,y∈G, x y=y x).  If the operation of G is commutative, G is an Abelian group (if not G is non-Abelian).

The order of G, written |G|, is the number of elements in G.

If a∈G, the order of a in G (unfortunately also written as |a|) is the smallest natural number n, if one exists, for which (a^n)=e.  If no such n exists we say |a|=∞.  Note than (a^(|a|))=e, but (a^k)=e does not necessarily mean |a|=k (as the order is the smallest positive power which turns a into e, not just some power).

If G is a group, a subset H of G is a subgroup if it is a group using the same operation as G.  For a subset H of G to be a subgroup it suffices that H contains the identity and that it is closed under products and inverses (i.e. e∈H and ∀x,y∈H, x y∈H∧(x^(-1))∈H).  If H is a subgroup of G we write H≤G.

A group G is called cyclic if there is an element a∈G for which every element g∈G can be written as a power of a (so ∀g∈G,∃n∈ℤ, g=(a^n)).  Such an a is called a generator of G, and the notation “G is cyclic and generated by a” is G=<a>.  Cyclic groups are always Abelian (and have other nice properties detailed below). Most groups are not cyclic.

Even if G is not cyclic, if a∈G then <a> = {(a^k)|k∈ℤ} is a subgroup called the cyclic subgroup generated by a. |<a>|=|a|. If |a|<∞, then (a^m)=(a^n) if and only if |a| | m-n.  This essentially says that |a|=k, then the exponent of a can be changed by multiples of k without changing the result (equivalently the exponent can be changed modulo k without changing the result).  A common application of this on exams is to compute high powers of an element whose order is known - for example if |a|=12 then (a^100)=(a^(100 mod 12))=(a^4).

The center of a group are those elements of G which commute with everything else in G (i.e. x is in the center of G if and only if ∀g∈G, g x =x g).  The center of G is written Z(G) and is a subgroup of G.  G is abelian if and only if G=Z(G).

If G is a group and a∈G, the centralizer of a in G, written C(a), is the set {x∈G|a x =x a} (the set of things that commute with a).  The center of the group is automatically a subgroup of C(a) regardless of what specific element a is.

Two groups (Subscript[G, 1]) and (Subscript[G, 2]) are called isomorphic if there is a function f:(Subscript[G, 1])->(Subscript[G, 2]) that is 1-1, onto, and operation-preserving (i.e. ∀x,y∈(Subscript[G, 1]), f(x y)=f(x)f(y)).  This is commonly written as (Subscript[G, 1])≅(Subscript[G, 2]).  Bring isomorphic is an equivalence relation, and it means the two groups are exactly the same (just using different notation) - everything from algebra that is true in one group is true in the other and any calculations in the two groups (such the number of elements of a given order) must have the same results.  An automorphism of G is an isomorphism of G with itself.  If a∈G, the inner automorphism induced by a is the function (Subscript[ϕ, a])(x)=a x (a^(-1)) (and as the name implies in an automorphism of G)

The Cayley table (sometimes called the operation table) for a group G is simply its multiplication table (typically with the identity written first).

Common Groups

The sets ℤ, ℚ, ℝ, and ℂ are all infinite groups under regular addition (identity is 0, the inverse of x is -x).  All of these groups are Abelian, and ℤ is cyclic (ℝ, ℚ, and ℂ are not cyclic).  ℤ=<1> = <-1>.  Every infinite cyclic group is isomorphic ℤ.

The sets (SuperStar[ℚ]), (SuperStar[ℝ]), and (SuperStar[ℂ]) (the non-zero elements of the given number systems) are groups under multiplication.  In ech case the identity is 1 and (x^(-1)) is really (1/x).  Sometimes a # symbol is used in place of * (so you may see notation like (ℝ^(#)), etc.)

The general linear group of n×n matrices with real entries (written (Subscript[GL, n])(ℝ)) which are those n×n matrices A with real entries whose determinant is not 0, is a group under matrix multiplication.   The identity is (Subscript[I, n]) and the inverse of A is (A^(-1)) (literally the inverse matrix of A).  The special linear group (Subscript[SL, n])(ℝ)={A∈(Subscript[GL, n])(ℝ)| det(A)=1} is a subgroup of (Subscript[GL, n])(ℝ).  Note that for the general and special linear groups the number system ℝ may be replaced by ℚ, ℂ, or (Subscript[ℤ, p]) (where p is prime).

For a natural number n, the group (Subscript[ℤ, n]) is the set {0,1,...,n-1} under addition modulo n.  The identity is 0, and the inverse of k is n-k (except for 0, which is its own inverse).  (Subscript[ℤ, n]) is Abelian and cyclic ((Subscript[ℤ, n])=<1>, although there usually other generators as well).  A cyclic group of order n is isomorphic to (Subscript[ℤ, n]).

For a natural number n≥1, the group of units modulo n, written U(n), is the set of numbers form 1 to n that are relatively prime to n (i.e. only have 1 as a common factor). under multiplication modulo n.  U(n) is Abelian but generally not cyclic (it is only cyclic for n=1,2,4,(p^k), and 2(p^k) where p is an odd prime).  The order of U(n) is the number of integers in the range 1,...n which are relatively prime to n.  This is commonly written as the “Euler ϕ function” ϕ(n).  The values of ϕ(n) can be quickly computed from the two rules if gcd(a,b)=1, ϕ(a b)=ϕ(a)ϕ(b) and for primes p we have  ϕ((p^k))=(p^k)-(p^(k-1)).  To compute the inverse of a∈U(n) you find a linear combination of a and n which is equal to 1 (that is, integers s and t for which a s+b n=1 - such combinations can be found by inspection or by the Euclidean Algorithm for computing gcd(a,n)=1) - (a^(-1))≡s mod n.

For a natural number n≥1, the complex (n^th) roots of unity are the solutions to the equation (z^n)=1 in ℂ.  The complex (n^th) roots of unity is a group of order n; the elements can be written as {cos(((360°)/n)k)+i sin(((360°)/n)k)|k=0,1,...,n-1}.  This group is cyclic for all n and has cos(((360°)/n))+i sin(((360°)/n)) as a generator.

For a natural number n≥3, (Subscript[D, n]) (the dihedral group of order 2n) is the group of symmetries of a regular n-gon under function composition (the identity is the rotation through 0 degrees (Subscript[R, 0])).  (Subscript[D, n]) is non-Abelian and therefore non-cyclic.  (Subscript[D, n]) consists of n rotations and n reflections (and so |(Subscript[D, n])|=2n.  The set of rotations is a cyclic subgroup of (Subscript[D, n]) of order n and is often written as <(Subscript[R, 360/n °])>.  The n reflections do not from a subgroup (as the identity (Subscript[R, 0]) is not a reflection), and if F is a reflection then |F|=2.  A rotation followed by a rotation is a rotation, a reflection followed by a reflection is a rotation, and a reflection and rotation (in either order) is a reflection.  If n is odd, Z((Subscript[D, n]))={(Subscript[R, 0])}; if n is even, Z((Subscript[D, n]))={(Subscript[R, 0]), (Subscript[R, 180])}. If H≤(Subscript[D, n]), then either every element of H is a rotation or the elements of H are half rotations and half reflections. Although the groups (Subscript[D, n]) are defined it terms of regular polygons the symmetries of a 2D shape are always a group, and many non-polygon shapes have symmetry groups which are isomorphic to (Subscript[D, n]); for example the three-leaved rose r=sin(3θ) has a symmetry group isomorphic to (Subscript[D, 3]).

PolarPlot[ Sin[3t], {t,0,2Pi}, Axes-> False]

The quaternions (often written as Q or (Subscript[Q, 8])) are a specific group of order 8.  Its elements are {1,-1,i,-i,j,-j,k,-k} with (i^2)=(j^2)=(k^2)=-1 and i j =k, j k=i, k i=j, j i =-k, k j=-i, and i k=-j.  It is a non-Abelian group (and therefore not cyclic).  Z((Subscript[Q, 8]))={-1,1}.

For n∈ℕ, the symmetric group of group of degree n, written (Subscript[S, n]), is the set of all permutations of the set {1,2,...,n} (functions which are from the set to itself which are 1-1, onto) under function composition (the identity ϵ is the identity function).  |(Subscript[S, n])|=n!.  The action of any permutation can be written as a combination of loops called disjoint cycle form. For example the permutation (1,2,3)(4,5) represents the action 1->2->3->1 together with 4->5->4 (many books do not use commas in cycle form if the range of numbers is understood from context).  In disjoint cycle form the identity is ϵ=(1).  The product of permutations α and β is commonly defined as αβ being the permutation where you plug numbers into β first and that result is plugged into α to get the final result.  In disjoint cycle form, (α^(-1)) is represented by the loops in the opposite direction - ((((1,2,3),(4,5,6,7)))^(-1))=(3,2,1)(7,6,5,4)=(1,3,2)(4,7,6,5).  If the cycles of α and β  have nothing in common, then αβ=βα.   (Subscript[S, n]) is non-Abelian for n≥3 (and therefore also not cyclic), and for n≥3 Z((Subscript[S, n]))={ϵ}.  In disjoint cycle form the order of a permutation is just the least common multiple of the cycle lengths.  A transposition/2-cycle in (Subscript[S, n]) is a permutation of the form (a b).  Every permutation in (Subscript[S, n]) is the product of 2-cycles; while this product is not unique, the evenness/oddness of the number of 2 cycles used in any decomposition is always the same.  This is used to define even and odd permutations (an even permutation is one with a breakdown into an even number of 2-cycles, and an odd permutation is one with a breakdown into an odd  number of 2-cycles).  No permutation is both even and odd, and the rules for multiplying even and off permutations are the same as for adding even and odd numbers.  The set of all even permutations form a subgroup of (Subscript[S, n]) called the alternating group of degree n and written (Subscript[A, n]). |(Subscript[A, n])|=((n!)/2) (so (Subscript[S, n]) is “half even and half odd”).  Every subgroup H of (Subscript[S, n]) is either all even or half-even and half-odd.  

Permutations can be defined on other sets rather than just {1,2,...n} - for example the elements in symmetry groups often act as permutations/rearrangements on the set of corners, the set of sides, etc.  If the sets of corners/sides/faces/etc. are finite with n elements, then by numbering the objects 1,2,...,n the symmetry group becomes isomorphic to a subgroup in (Subscript[S, n]).  If a group G acts as a group of permutations on a set S, then for any i∈S the stabilizer of i under G is the set (Subscript[stab, G])(i)={α∈G|α(i)=i} (the stabilizer of i are those permutations which leave i alone).  The stabilizer is always a subgroup of G.  The orbit of i under G is the set (Subscript[orb, i])(G)={α(i)|α∈G}, which is the set of places i gets sent to under the permutations in G/

If (Subscript[G, 1]), (Subscript[G, 2]),..., (Subscript[G, n]) are groups, the external direct product of (Subscript[G, 1]), (Subscript[G, 2]),..., (Subscript[G, n]), written (Subscript[G, 1])⊕(Subscript[G, 2])⊕...⊕(Subscript[G, n]) is the set of n-tuples ((Subscript[g, 1]),(Subscript[g, 2]),...,(Subscript[g, n])) where (Subscript[g, k])∈(Subscript[G, k]) (this can also be written using ⊗ - the difference between ⊕ [formally “external direct sum”] and ⊗ [“external direct product”] only comes up when you have an infinite set of groups, which is unlikely to come up on a timed exam).  This is a group under the component-wise operation - essentially this is “vectors from groups”.  The identity is the n-tuple ((Subscript[e, 1]),(Subscript[e, 2]),...,(Subscript[e, n])) made from the identities of the groups and the inverse of ((Subscript[g, 1]),(Subscript[g, 2]),...,(Subscript[g, n])) is (((Subscript[g, 1])^(-1)),((Subscript[g, 2])^(-1)),...,((Subscript[g, n])^(-1))).  The order of  (Subscript[G, 1])⊕(Subscript[G, 2])⊕...⊕(Subscript[G, n]) is |(Subscript[G, 1])|×|(Subscript[G, 2])|×...×|(Subscript[G, n])|.  (Subscript[G, 1])⊕(Subscript[G, 2])⊕...⊕(Subscript[G, n]) is Abelian if and only if all the individual groups are Abelian, and (Subscript[G, 1])⊕(Subscript[G, 2])⊕...⊕(Subscript[G, n]) is cyclic if and only if all the groups are cyclic and the orders |(Subscript[G, k])| are pairwise relatively prime.

If G is a group, the set of all automorphisms of G is a group (written Aut(G)).  The set of all inner automorphisms of G is also a group (written Inn(G); if Z(G) is finite then |Inn(G)|=((|G|)/(|Z(G)|)).

Groups of Low Order/Specific Forms

Here is a list of groups (up to isomorphism) of order up to 10 together with some information about groups whose order has a particular form:

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Given a group of a given size you can sometimes use this table to determine what the given group is isomorphic to by process of elimination.  The key qualities often used in process of elimination are cyclic/non-cyclic, Abelian/non-Abelian, and the number of elements of a given order.

Cosets and Normal Subgroups

Let H be a subgroup of a group G.  For any a∈G, the left coset for H containing a is the set a H={a h|h∈H} (right cosets H a are defined in a similar way but are rarely used).  Properties of cosets include:

a∈a H

| a H | = | H |

For any a,b∈G, either a H=b H or a H⋂b H=∅

If b∈a H, b H=a H.

If G is a finite group, the number of distinct left cosets of H  in G (called the index of H in G, written as |G:H| or [G:H]) is equal to |G|/|H|.

a H=H a if and only if ∀h∈H, a h (a^(-1))∈H

A subgroup Hof G is called a normal subgroup (written H⊲G) if for all a∈G, a H=H a .  To show a subgroup H is normal it suffices to show ∀a∈G,∀h∈H, a h (a^(-1))∈H.  Examples of normal subgroups are:

all subgroups of an Abelian subgroup

The center Z(G) of any group G

the full set of rotations in (Subscript[D, n])

(Subscript[A, n]) in (Subscript[S, n])

any subgroup H of a finite group G for which |H|=((|G|)/2) (this includes the previous two examples).

If H is normal in G the set of all left cosets can be turned into a group using the operation a H b H=(a b)H.  This is called a factor group or quotient group and typically written G/H.  If |G| is finite, |G/H|=|G|/|H|.  The difficulty in working with this operation is the same left coset can typically be written as a H for multiple different group elements a (a consequence of the property that if b∈a H, b H=a H).  So in  making an operation/Cayley table for G/H you typically have to choose a “standard” way to write each left coset (choosing a specific a for the a H) and when using  a H b H=(a b)H you  may  have to “convert” the result (a b)H to the “standard name” for that coset.

Important Theorems About Groups

Cyclic groups

Let G=<a> be a cyclic group of order n.  Then:

All subgroups of G are cyclic

For each divisor k of n there is exactly one subgroup H  in G or order k - namely <(a^(n/k))>

The number of elements in G with order k is ({{{, ({{0, k∤n}, {ϕ(k), k|n}})}})

The set of all generators of G is found by taking a known generator a and raising it to the elements of the group U(n) (i.e. numbers from 1 to n relatively prime to n)

(Subscript[ℤ, m])⊕(Subscript[ℤ, n])≅(Subscript[ℤ, m n]) if and only if m and n are relatively prime (i.e. gcd(m,n)=1)

U(2)≅(Subscript[ℤ, 1]).  U(4)≅(Subscript[ℤ, 2]).  For  k≥3, U((2^k))≅(Subscript[ℤ, 2])⊕(Subscript[ℤ, (2^(k-2))]).  If p is an odd prime, U((p^k))≅(Subscript[ℤ, (p^k)-(p^(k-1))]).  Together with the fact that if gcd(a,b)=1, U(a b)≅U(a)⊕U(b), this is often used to write the group U(n) (for a specific n) as a product of cyclic groups.

The previous theorem can be interpreted as a set of rules for how to compute values of ϕ(n) for n≥2

If a and b are relatively prime, ϕ(a b)=ϕ(a)ϕ(b)

ϕ((2^k))=(2^(k-1))=(1/2)×(2^k) for k∈ℕ

If p is an odd prime, ϕ((p^k))=(p^k)-(p^(k-1)) for k∈ℕ (this actually works for p=2, but most people find the “take half” rule above to be easier to use).

Orders of Elements and Subgroups

If G is a finite group and H≤G, |H| divides |G|  (Lagrange’s theorem)

All groups of prime order p are cyclic (and therefore isomorphic to (Subscript[ℤ, p]))

If G is a finite group and a∈G, |a| divides |G| (note: G may not have any elements whose order is a given factor unless G is cyclic)

If p is a prime number and p| |G|, G will have an element of order p.

If p is a prime and (p^k)| |G|, then G has a subgroup of order (p^k).  If (p^k)| |G| but (p^(k+1))∤ |G| (i.e. (p^k) exactly divides |G|), a subgroup of order (p^k) is called a Sylow p-subgroup of G.  If (Subscript[n, p]) is the number of different Sylow p-subgroups of G, (Subscript[n, p])≡1 mod p and (Subscript[n, p])| (|G|/(p^k)), where (p^k) exactly divides |G|.  A unique p-Sylow ((Subscript[n, p])=1) is always normal in G.

If p is a prime number, then for any number a, (a^p)≡a mod p; if p∤a, then (a^(p-1))≡1 mod p (this is Fermat’s Little Theorem, which is essentially Lagrange’s Theorem applied to U(p)).

If p is a prime and G is a group with |G|=(p^k), then Z(G)≠{e} (i.e. the center has more than one element).

If H and K are finite subgroups of G, then H⋂K is a subgroup of G whose order divides both |H| and |K|.

Orbit-Stabilizer Theorem

Let G be a finite group of permutations on a set S.  Then for any i∈S, |G|=|(Subscript[stab, G])(i)|×|(Subscript[orb, G])(i)|.  This theorem is commonly used to determine the size of a symmetry group by thinking of it as a group of permutations.  Pick a convenient part i of the object, count the number of symmetries that leave it alone, the number of places it gets moved to, and multiply them together to the size of the group.  If you did not already know the symmetries of the square (Subscript[D, 4]) had 8 elements, you could track the top right corner.  The symmetries that leave it in place would be (Subscript[R, 0°]) and the flip about the main diagonal (so |stab|=2); the symmetries take the top left corner to all 4 corners (so |orb|=4).  So |(Subscript[D, 4])|=|stab|×|orb|=2 4=8.

Fundamental Theorem of Finite Abelian Groups

If G is a finite Abelian group of order n, it is isomorphic to an external direct product of cyclic groups of prime-power order (i.e. groups (Subscript[ℤ, (p^k)]) for some prime p). The primes p involved are exactly those that that divide n, and for a given prime p the product of all the prime powers (p^k) used in the product give the prime power (p^m) that exactly divides n.

To find all the Abelian groups of a given order n (up to isomorphism):

Find the prime factorization of n as ((Subscript[p, 1])^(Subscript[k, 1]))((Subscript[p, 2])^(Subscript[k, 2]))((Subscript[p, 3])^(Subscript[k, 3]))...

For each individual power ((Subscript[p, j])^(Subscript[k, j])), find all the ways to group powers of (Subscript[p, j]) together that when multiplied give ((Subscript[p, j])^(Subscript[k, j])) (for example, (5^3)=125 could come from (Subscript[ℤ, 125]), (Subscript[ℤ, 25])⊕(Subscript[ℤ, 5]), and (Subscript[ℤ, 5])⊕(Subscript[ℤ, 5])⊕(Subscript[ℤ, 5])).  Order does not matter in these groupings.

To list all Abelian groups of order n, for each prime p in the factorization choose one breakdown from the previous step and join them with the external direct product.  The complete set of such choices leads to all the possible groups.

For example to find all Abelian groups of order 72:

72=(2^3)(3^2)

(2^3)=8 can be broken down as (Subscript[ℤ, 8]), (Subscript[ℤ, 4])⊕(Subscript[ℤ, 2]), and (Subscript[ℤ, 2])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 2]); (3^2)=9 can be broken down as (Subscript[ℤ, 9]) or (Subscript[ℤ, 3])⊕(Subscript[ℤ, 3]).

Taking all combinations of the “power of 2’s” breakdown with all the options in the “powers of 3’s” breakdowns, we have: (Subscript[ℤ, 8])⊕(Subscript[ℤ, 9]), (Subscript[ℤ, 8])⊕(Subscript[ℤ, 3])⊕(Subscript[ℤ, 3]), (Subscript[ℤ, 4])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 9]),  (Subscript[ℤ, 4])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 3])⊕(Subscript[ℤ, 3]), (Subscript[ℤ, 2])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 9]), and (Subscript[ℤ, 2])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 3])⊕(Subscript[ℤ, 3]).

These combinations are all distinct (up to isomorphism), and are sometimes written in a standard form by combining the highest prime powers from each prime component together (and repeating) using the fact that (Subscript[ℤ, m])⊕(Subscript[ℤ, n])≅(Subscript[ℤ, m n]) if and only if m and n are relatively prime.  So in the same order as before these are (Subscript[ℤ, 72]), (Subscript[ℤ, 24])⊕(Subscript[ℤ, 3]), (Subscript[ℤ, 36])⊕(Subscript[ℤ, 2]), (Subscript[ℤ, 12])⊕(Subscript[ℤ, 6]), (Subscript[ℤ, 18])⊕(Subscript[ℤ, 2])⊕(Subscript[ℤ, 3]), and (Subscript[ℤ, 6])⊕(Subscript[ℤ, 6])⊕(Subscript[ℤ, 2]).

Group Homomorphisms

If (Subscript[G, 1]) and (Subscript[G, 2]) are groups, a group homomorphism from (Subscript[G, 1]) to (Subscript[G, 2]) is a function α:(Subscript[G, 1])->(Subscript[G, 2]) which is operation preserving (that is, ∀x,y∈(Subscript[G, 1]), α(x y)=α(x)α(y).  Homomorphisms need not be 1-1 or onto (which are the extra requirements to be an isomorphism). If α is a homomorphism then the kernel of α is the set ker(α)={x∈(Subscript[G, 1])|α(x)=(Subscript[e, 2])} where (Subscript[e, 2]) is the identity of (Subscript[G, 2]).  The kernel of α is never empty as (Subscript[e, 1])∈ker(α), where (Subscript[e, 1]) is the identity of (Subscript[G, 1]).

Some examples of homomorphisms are:

α:ℤ->ℤ given by α(x)=2x.  In this case ker(α)={0}.

α:ℤ->(Subscript[ℤ, n]) given by α(x)=x mod n.  In this case ker(α)=<n>.

For any natural number n, α:(SuperStar[ℂ])->(SuperStar[ℂ]) given by α(z)=(z^k) is a homomorphism.  The kernel of α is the (k^th) roots of unity.

α: (SuperStar[ℝ])->ℝ given by α(x)=ln(x) is a homomorphism with kernel {1}.

For any groups (Subscript[G, 1]) and (Subscript[G, 2]), α:(Subscript[G, 1])⊕(Subscript[G, 2])->(Subscript[G, 1]) given by α((x,y))=x is a homomorphism and ker(α)={((Subscript[e, 1]),y)|y∈(Subscript[G, 2])}.  The map  β((x,y))=y is also a homomorphism.

Homomorphisms have many important properties.  If α:(Subscript[G, 1])->(Subscript[G, 2]) is a homomorphism, then some of these properties are:

ker(α) is a normal subgroup of (Subscript[G, 1]).

If |ker(α)|=n, then α is an n-to-1 map (so showing the kernel is just {(Subscript[e, 1])} is a way to show α is 1-1).

|α(x)| | |x|.

The range of α is a subgroup of (Subscript[G, 2]).  If |ker(α)|=n<∞, then the size of the range is |(Subscript[G, 1])|/n.  In fact the range of α is isomorphic to the factor group (Subscript[G, 1])/ker(α).

If α(x)=y, then the pre-image (α^(-1))(y)={z∈(Subscript[G, 1])|α(z)=y} is equal to the left coset x ker(α).

Rings, Integral Domains, and Fields

A ring R is a set with two binary operations (one written as addition and another written as multiplication) for which

There is an additive identity 0.

Every element x has an additive inverse -x

Multiplication is associative and addition is both associative and commutative (so x+y=y+x)

Multiplication left- and right-distributes over addition.

Note that the multiplication does not have to be commutative (if it is, R is a commutative ring) nor have a multiplicative identity (if it does it is called the unity, is often written as 1 or (Subscript[1, R]), and R is a ring with unity).  

In a commutative ring with unity, elements that have a multiplicative inverse are called units (the set of all units in such a ring is an Abelian group called the group of units).

In a commutative ring with unity, two elements a and b are called associates if and only if there is a unit u with a=u b (which would also imply b=(u^(-1))a).

A subring of R is a non-empty subset of R which is a ring itself with the same operations.  Subrings do not play quite as important role for rings as subgroups do for groups.

An ideal of R is a subring I of R which has the extra property that ∀a∈I, ∀r∈R, r a∈I and a r∈I.  So ideals are not only closed under multiplication, they are also closed under multiplication from the full ring (think of even numbers - while is true that even × even=even, it also true that any integer × even = even).  Ideals are the ring-theory analog of normal subgroups and do play a large role in ring theory (including the notation of the factor ring R/I, which is defined the same way as a factor group but you have (a+I) + (b+I)= (a+b)+I and (a+I)(b+I)=a b+I).

If R is a commutative ring with unity and x∈R, then the principal ideal generated by x is the set <x> ={ r x|r∈R} (some books use the notation (x) in place of <x>).  An ideal I is called principal if I=<x> for some x∈R

Non-zero ring elements a and b are called zero-divisors if a b=0.

A ring element x  is called nilpotent if (x^n)=0 for some n∈ℕ.

A ring element x is called idempotent if (x^2)=x.

In a commutative ring with unity, an ideal I is called a prime ideal if a b∈I implies either a∈I or b∈I.

In a commutative ring with unity, a proper ideal I is called a maximal ideal if the only ideal J which properly contains I is the entire ring itself.  A maximal ideal is automatically a prime ideal.

An integral domain is a commutative ring with unity with no zero-divisors (so in such a ring a b=0 implies a=0 or b=0).

A field is a commutative ring with unity in which all non-zero elements have multiplicative inverses (i.e. every non-zero element is a unit).  Fields are commonly used in linear algebra (in fact for elementary linear algebra real numbers can be replaced with any field F and everything still works).  A field is automatically an integral domain.

An integral domain is called a Euclidean domain if it is equipped with a notion of the “size of a non-zero element” and a corresponding division algorithm.

Examples of Rings and Ideals

The sets ℤ, ℚ, ℝ, and ℂ are all rings under their usual additions and multiplications.  ℤ is an integral domain but not a field; the units of ℤ are just {1,-1}.  ℚ, ℝ, and ℂ are all fields.  ℤ is a principal ideal domain and a Euclidean domain (the size of a non-zero number k is just |k| and the division algorithm is just long division).

The Gaussian integers ℤ[i]={a+b i| a,b∈ℤ} are a commutative ring with unity.  In addition ℤ[i] is both an integral domain and a Euclidean domain (with “size” of an element given by absolute value; the quotient in long division of a+b i by c+d i is found by writing ((a+b i)/(c+d i)) in standard form and rounding the real and imaginary parts).

The set (Subscript[M, n])(ℝ) (the set of n×n matrices with real entries) is a ring using matrix addition and multiplication.  It is non-commutative but has an identity ((Subscript[I, n])).

The set (Subscript[ℤ, n]) is a commutative ring with unity under addition and multiplication mod n.  (Subscript[ℤ, n]) will have zero-divisors if nis not prime; if n is prime it is both an integral domain and field.  The group of units of (Subscript[ℤ, n]) is U(n) (which is where the name of U(n) really comes from).  Every ideal in (Subscript[ℤ, n]) is principal (so (Subscript[ℤ, n]) is sometimes called a “principal ideal ring”).  Depending on the value of n the ring (Subscript[ℤ, n]) may or may not have nilpotent elements other than 0 (0 is always nilpotent) or idempotents other than 0 and 1 (which are always idempotent).  In (Subscript[ℤ, 5]) the only nilpotent element is 0 and the only idempotent element is 1.  In (Subscript[ℤ, 12]) however 6 is nilpotent (as (6^2)≡36≡0) and 9 is idempotent (as (9^2)≡(((-3))^2)≡9).

The set ℝ[x] is the set of all polynomials with real coefficients.  It is a principal ideal domain but not a field.  The units of ℝ[x] are the non-zero constants.  ℝ[x] is a Euclidean domain (with “size” given by degree of a polynomial and the division algorithm is polynomial long division).  ℝ can be replaced here by ℚ, ℂ, (Subscript[ℤ, p]), or any field and these statements will still be true).

The set of continuous functions on an interval [a,b] (or all of ℝ) is a ring under normal function addition and multiplication.  It is not an integral domain.  For example if f(x)=({{{, ({{x, x>0}, {0, x≤0}})}})  and g(x)=({{{, ({{0, x≥0}, {x, x<0}})}}), then f(x)g(x)=0 even though neither f(x) nor g(x) is the zero function.

In ℤ, as it is a principal ideal domain every ideal has the form <k> = k ℤ for some k≥0.  The prime ideals are exactly those of the form <0> = {0} and<p>, where p is a prime number.  The only maximal ideals are those <p>, where p is a prime number (so the ideal <0> is prime but not maximal as <0> is a proper subset of the ideal <2>).

If (Subscript[R, 1]), (Subscript[R, 2]),..., (Subscript[R, n]) are rings, the external direct product (Subscript[R, 1])⊕(Subscript[R, 2])⊕...⊕(Subscript[R, n]) is a ring under component-wise addition and multiplication.

If (Subscript[R, 1]) and (Subscript[R, 2]) are commutative rings with unity, so is (Subscript[R, 1])⊕(Subscript[R, 2]) (the unity is the ordered pair ((Subscript[1, (Subscript[R, 1])]),(Subscript[1, (Subscript[R, 2])]))).  The set I={(x,(Subscript[0, (Subscript[R, 2])])}|x∈(Subscript[R, 1])} is an ideal of R (as the the similar set defined using points with first coordinate 0).  Note that even if (Subscript[R, 1]) and (Subscript[R, 2]) are integral domains (Subscript[R, 1])⊕(Subscript[R, 2]) is not an integral domain - ((Subscript[0, (Subscript[R, 1])]),(Subscript[1, (Subscript[R, 2])]))×((Subscript[1, (Subscript[R, 1])]),(Subscript[0, (Subscript[R, 2])]))=((Subscript[0, (Subscript[R, 1])])(Subscript[1, (Subscript[R, 1])]),(Subscript[1, (Subscript[R, 2])])(Subscript[0, (Subscript[R, 2])]))=((Subscript[0, (Subscript[R, 1])]),(Subscript[0, (Subscript[R, 2])])) even though neither starting pair is the 0 element ((Subscript[0, (Subscript[R, 1])]),(Subscript[0, (Subscript[R, 2])])).

In a commutative ring with unity R the set N of all nilpotent elements is an ideal (with closure under addition/subtraction following from the binomial theorem).  This ideal is called the nilradical of R.

Ring Homomorphisms

If R and S are rings, a ring homomorphism from R to S  is a function α from R to S which preserves both addition and multiplication - that is ∀x,y∈R, α(x+y)=α(x+y) and α(x y)=α(x)α(y).  If R and S both have a unity, many books would also require that α((Subscript[1, R]))=(Subscript[1, S]) (this is not universally accepted).  If α is a ring homomorphism from R to S, the kernel of α is defined the same way as it is for groups - ker(α)={x∈R| α(x)=(Subscript[0, S])}.  Some examples of ring homomorphisms are:

The function α:ℤ->(Subscript[ℤ, n]) given by α(x)=x mod n is a homomorphism with ker(α)=<n>.

The function α:ℤ[i]->ℤ[i] given by α(a+b i)=a-b i is a homomorphism with ker(α)={0}.

The function α:ℝ[x]->ℝ given by α(p(x))=p(7) is a homomorphism with ker(α)=<x-7>.

If R and S are rings, the function α:R⊕S->R given by α((x,y))=x is a homomorphism with ker(α)={((Subscript[0, R]),s)|s∈S}

Ring homomorphisms have many important properties, many of which follow from the fact that they are also group homomorphisms (if you just ignore all the multiplication properties).  If α:R->S is a ring homomorphism, then:

ker(α) is a ideal of R.

If |ker(α)|=n, then α is an n-to-1 map (so showing the kernel is just {(Subscript[0, R])} is a way to show α is 1-1).

The range of α is a subring of S.  If |ker(α)|=n<∞, then the size of the range is |R|/n.  In fact the range of α is isomorphic to the factor ring (Subscript[G, 1])/ker(α).

If α(r)=s, then the pre-image (α^(-1))(s)={r∈R|α(r)=s} is equal to the left coset r ker(α).

There are many other properties of ring homomorphisms but these are the ones most likely to come up on a timed exam.

A ring isomorphism is a ring homomorphism which is both 1-1 and onto.  As with group isomorphisms, if there is an isomorphism between two rings they are considered the same.

Important Ring Theorems

In a commutative ring with unity every maximal ideal is also prime.

Every Euclidean domain is also a principal ideal domain (that is in a Euclidean domain every ideal has the form<x> for some x).

In a commutative ring with unity, an ideal I is prime if and only if the factor ring R/I is an integral domain.

In a commutative ring with unity, and ideal I is maximal if and only if the factor ring R/I is a field. (this is why maximal ideals are prime; every field is also an integral domain).

If α:R->S is a ring homomorphism, the range of α is ring isomorphic to the factor ring R/ker(α).

Every finite integral domain is a field.