Ch. 3 Finite Groups and Subgroups

Question	Answer
Define: Order of a Group	The number of elements of a group is called its order, or \|G\|.
Order of an element	The order of an element g in a group G is the smallest positive integer n such that g^n = e. If no such integer exists, then g as infinite order, or \|g\| = infinity
Subgroup	If a subset H of a group is itself a group under the operation of g, we say that H is a subgroup of G.
One-Step Subgroup Test	Consider H, a nonempty subset of G. If ab^(-1) is in H for every a,b in H, then H is a subgroup of G.
Two-Step Subgroup Test	Consider H, a nonempty subset of G. If: 1.) every element of H has an inverse, and 2.) ab is in H for every a,b in H then H is a subgroup of G
Finite Subgroup Test	Let H be a nonempty finite subset of a group G. If H is closed under operation G, then H is a subgroup of G.
<a> is a subgroup	Let G be a group, and let a be any element of G. Then <a> is a subgroup of G.
Define: Center of a Group	The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. The center is a subgroup of G.
Define: Centralizer	Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. C(a) is a subgroup of G.

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	Created by William Hartemink about 7 years ago