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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