Basic Properties of Groups

Question 1

Group

Answer 1

A set (G,x) [{Set, operation}] s.t.:

(0) [closure]: for all a, b in G, ab in G

(1) [associative]: for all a,b,c in G, (ab)c = a(bc)

(2) [identity]: there exists e in G s.t. for every a in G, ae=a

(3) [inverse]: for every a, there exists b in G s.t. ab=e

Question 2

Order of a Group

Answer 2

• denoted |G|
• number of elements in the group (could be infinite)

Question 3

Order of an element

Answer 3

• denoted |g|
• smallest positive integer n s.t. gⁿ =e (if no such n exists, we say g has infinite order)

Question 4

Important Groups

Answer 4

• D_n (dihedral group of order 2n, symmetries of regular n-gon)
• Z_n (integers mod n, under add)
• U(n) (units mod n, multiplication)

Question 5

Cayley Tables

Answer 5

• “multiplication” tables for groups
• if symmetric, then abelian group

Question 6

Subgroup

Answer 6

A non-empty subset, H, of a group, G, if H is also a group under the same operation as G

{e} is the trivial subgroup

Question 7

Two-step Subgroup Test

Answer 7

Let H be a subset of G. Suppose
0. H is non-empty
1. a,b in H implies ab in H
2. a in H implies a^-1 in H

Question 8

One-step Subgroup Test

Answer 8

Let H be a subset of G. Suppose
0. H is non-empty
1. a,b in H implies ab^-1 in H

Question 9

Cyclic Subgroup Generated by an Element

Answer 9

• denoted ⟨a⟩
• subgroup {aⁿ in G | n in N }
• Note: G is cyclic if G = ⟨a⟩ for some element a in G

Question 10

Center of a Group

Answer 10

• Z(G) = {b in G | ab=ba for every a in G}
• b in Z(G) iff b commutes w/ all elements in G

Question 11

Centralizer of an Element

Answer 11

Centralizer of a in G is
C(a) = {b in G | ab=ba}
i.e. C(R₉₀) = {R₀, R₉₀, R₁₈₀, R₂₇₀}