Skip to main content Link Search Menu Expand Document (external link) Copy Copied

Quick Review of Group Theory

Key definitions

Definition: A group is a set G together with a map m:G×GG satisfying the following axioms:

  1. There is an element eG such that m(e,x)=m(x,e)=x for all xG.
  2. For all x,y,zG, we have m(x,m(y,z))=m(m(x,y),z).
  3. For all xG, there is yG such that m(x,y)=m(y,x)=e.

We usually just write ab or a+b for m(a,b); and we usually write G, rather than (G,m) when speaking about a group.

One can weaken these axioms in various ways and obtain an equivalent definition.

For any group G and xG:

  • there is only one element e satisfying axiom (1).
  • the regrouping in axiom (2) extends to arbitrary many elements, so the product a1a2an is well defined for any set of elements a1,a2,,anG.
  • the inverse y for x required by axiom (3) is unique.

Definition: If G is a group and ab=ba for all a,bG then G is called abelian.

Definition: If G is a group and gG then the order of g is the smallest positive integer n such that gn=e (or infinity, if no such n exists).

Definition: If G is a group, and H is a nonempty subset of G which is a group when the multiplication of G is restricted to H, then H is called a subgroup of G. H is a subgroup if:

  • for all a,bH, abH,
  • if aH, then a1H.

Definition: If G and H are groups, and f:GH is a function, then f is called a homomorphism if f(ab)=f(a)f(b) for all a,bG and and isomorphism if it is a bijective homomorphism. The kernel of a homomorphism f is the subgroup of G consisting of elements g such that f(g)=e. The image of a homomorphism f is the subgroup of H consisting of elements hH such that h=f(g) for some gG.

Definition: If G is a group and X is a set, then a map m:G×XX is called a (left) action of G on X if m(e,x)=x for all xX and m(a,m(b,x))=m(ab,x) for all a,bG and xX. We usually write ax for m(a,x).

Examples

  1. The integers, rational numbers, real numbers, and complex numbers are all groups under addition.
  2. The nonzero rational numbers, real numbers, and complex numbers are all groups under multiplication.
  3. For n>0, the set Z/nZ of integers modulo n are a group under addition.
  4. The subset of Z/nZ consisting of elements a that are invertible (i.e. such that the congruence ax1(modn) has a solution) form a group under multiplication. This group is called (Z/nZ)×. If n=p is prime, then (Z/pZ) consists of the p1 nonzero congruence classes.
  5. The invertible n×n matrices GLn(F) where F is any of Q,R,C form a group under matrix multiplication. These groups come with actions on Fn given by matrix multiplication.
  6. For any set X, the set of bijective maps XX form a group under composition of functions. This group is called the symmetric group S(X) on X. If there is a bijection from X to Y, then S(X) and S(Y) are isomorphic. If X={1,2,3,,n} then S(X) is usually written Sn and is called the symmetric group on n elements. Notice that S(X) comes with an action on X given by (f,x)f(x).
  7. If X is a regular ngon in the plane, the group of rigid motions f of the plane such that f(X)=X form a group under composition called the Dihedral group. Dummit and Foote call this group D2n since it has 2n elements, but others call it Dn since it is the symmetries of an n-gon. The elements of D2n consist of {1,r,r2,,rn1,s,sr,sr2,,srn1} and the group law is determined by the rules rirj=ri+j with exponents read modulo n, s2=1, and sr=r1s. The group D2n comes with an action on the n vertices of X by (f,v)f(v) and on the n sides of v by (f,s)=f(s).

Cyclic Groups

Definition: A group G is cyclic if there is an element gG such that the homomorphism ϕg:ZG defined by ϕg(n)=gn is surjective.

Proposition: Let HZ be a propersubgroup. Then either H=0 or there is a unique n>0 such that H=nZ.

Corollary: A cyclic group is isomorphic either to Z or to Z/nZ for some integer n>0.

Properties of order: Let G be a group and xG.

  1. If xa has infinite order for some a, so do all nonzero powers of x.
  2. If xa=e and xb=e then xgcd(a,b)=e.
  3. If xa=e then the order of x divides a.
  4. If x has order a, then xk has order a/gcd(a,k).
  5. If G is cyclic of order n generated by x, then xa generates G if and only if gcd(a,n)=1.

Proposition: The subgroups of G=Z/nZ are in bijection with the divisors of n. If d is a divisor of n, then the unique subgroup of G of order d is generated by n/d.

Euclid’s Algorithm and Congruences

Theorem: Let a and b be nonzero integers. Then there exist integers x and y such that ax+by=d where d is the greatest common divisor of a and b.

Theorem: Let n be a positive integer. The congruence equation

axb(modn)

has solutions if and only if d=gcd(a,n) divides b. If this condition is satisfied, it has d solutions of the form

x0+kndk=0,,d1

where x0 is a representative for the unique solution to the congruence

adx0bd(modnd).

Remark: Notice that the congruence equation problem is equivalent to finding x and y so that ax+ny=b.