This preview shows page 1. Sign up to view the full content.
Definitions for Geometric Groups
Definition: Let G be a nonempty set with a binary operation *.
G is a group under *
in case:
(i)
If a, b are elements of G, then a*b is an element of G. (Closure)
(ii)
If a, b, c are any three elements of G, then (a*b)*c = a*(b*c). (Associative)
(iii)
There exists an identity element e in G such that g*e = e*g = g for any element g in
G. (Identity)
(iv)
If g is an element of G, then there exists an inverse element for g, denoted g
–1
, such
that
g * g
–1
= g
–1
* g = e. (Inverse property).
(Examples: {all integers} under addition, {all nonzero fractions} under multiplication, {all
symmetries of a polygon in the plane} under “followed by”
Definition: Let G be a group with binary operation * and H a nonempty subset of G. Then H is a
subgroup
of G if H is also a group under the operation *.
Example: { all multiples of 2} is a subgroup of {all integers} if the operation is addition.
{all multiples of 2} isn’t a subgroup of {all integers} under addition, if the operation
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 05/11/2008 for the course MATH 101 taught by Professor Dechene during the Spring '08 term at Fitchburg.
 Spring '08
 Dechene

Click to edit the document details