Abstract Algebra Exam 1

Front 1

Abelian Group

Back 1

A group is abelian if ab = ba for all a,b in G

Front 2

cyclic subgroup generated by an element

Back 2

if G is a group and a is in G, then the set <a> = { a^n I n is an integer} is called the cyclic subgroup of G generated by a.

Front 3

One Step Supgroup Test

Back 3

Let G be a group and H be a nonempty subset of G. If ab^-1 is in H whenever a and b are in H then H is a subgroup of G
0. nonempty
1. If a, b in H prove ab^-1 in H

Front 4

Two Step Supgroup Test

Back 4

Let G be a group and H be a nonempty subset of G. If ab is in H whenever a,b are in H and a^-1 is in H whenever a is in H, then H is a subgroup of G.
0. nonempty
1. Closed
2. Inverse

Front 5

Centralizer of the element a : C(a)

Back 5

is the set of all elements in G that commute with a

Front 6

Center of a group: Z(G)

Back 6

{ x in G I xg = gx for all g in G}

Front 7

Supgroup

Back 7

if a subgroup H of a group G is itself a group under the operation of G, we say that H is a subgroup of G.

Front 8

Order of an element in a group

Back 8

is the smallest positive integer such that g^n = e

Front 9

Order of a group

Back 9

the number of elements, infinite or finite

Front 10

t divides s

Back 10

t is a divisor of s if s = tm for some integer m