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10 Cards in this Set
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Abelian Group

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


cyclic subgroup generated by an element

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.


One Step Supgroup Test

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 

Two Step Supgroup Test

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 

Centralizer of the element a : C(a)

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


Center of a group: Z(G)

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


Supgroup

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.


Order of an element in a group

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


Order of a group

the number of elements, infinite or finite


t divides s

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