External Direct Product
Let Gi, Gz, . . . , G" be a finite collection of groups. The external direct
product of G], Gz, . . . , G, written as Gl Q Gr2 Q ° ' - Q G", is the set of
all n-tuples for which the ith component is an element of GI. and the
opera
Chapter 2: Groups
Definition Binary Operation
Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G.
Definition Group
Let G be a set together with a binary operation (usually called multiplic
Math 430-001 (Ellis)
Chapter 6: Isomorphisms.
Fall 2008
Theorem 6.2 Properties of Isomorphisms Acting on Elements.
Let G, G be groups with respective identities e, e. Let k, n Z and a, b G.
Then
1. (e) = e;
2. (an ) = [(a)]n ;
3. ab = ba i (a)(b) = (b)(a)
Exam 1: Core definitions
Should memorize:
Definition Group
Let G be a set together with a binary operation (usually called
multiplication) that assigns to each ordered pair (a, b) of elements of
G an element in G denoted by ab. We say G is a group under t
Motion:
R0: Rotation of 0o (no change in position)
R90: Rotation of 90o
R180: Rotation of 180o
R270: Rotation of 270o
H: flip about a horizontal axis
V: flip about a vertical axis
D: flip about the main diagonal
D: flip about the other diagonal
Group:
The