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Algebraic Structures Definitions
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Key Concepts:
Terms in this set (14)
Principle of well ordering
Every non-empty subset of the natural numbers has at least one element
Division Algorithm
If alb are integers then there are unique integers q and r with 0 <= r <b so that a=qb+r (if a<b, q=0)
Greatest Common Divisor
If a,b are integers gcd(a,b) = ar + bs for some r,s that are integers
Binary Operation
An operation that takes two inputs and gives one output
Group
(G,
) is a set G together with a binary operation
:GxG -> G
1) There is an identity in the group, for all g in G, g
e=e
g=g
2) For every g in G, there is an inverse element g^-1 in G so that gg^-1=g^-1g=e
3) For all g,h,k in G g
(h
k)=(g
h)
k
Group
(G,
) is a set G together with the operation
:GxG-> G such that
1) G has an identity with respect to *
2) Every element of G has an inverse with respect to *
3)* is associative
Abelian
A group is called if for every g,h in G, gh=hg
Order
of a group is the number of elements in the group. ord(G)=|G|
Subgroup
H, is a subset H of G, if H is a group with respect to the operation of G.
Proper Subgroup
a subgroup of a group G that is not G or {e} is called
Cyclic subgroup
for a in G we call <a> the generated by a
cyclic group
If G contains some element 'a' such that G=<a>, then G is a
generator
when G=<a>, then a is a of G
Order
If 'a' is an element of G, we define the of 'a' to be the smallest positive integer n such that a^n=e, and we write |a| = n. If there is no such integer n, we say that the order of 'a' is infinite and write |a| = infinity to denote the order of a.
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