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Abstract Definitions

binary operation let A be a set . A binary operation * on A is a function *: A --> A (a,b) |--->a*b that is given any two elements a,b in A. there is a unique a*b in A
Closed if for each a,b in A we have a*b in A we say A is closed under the operation *
algebraic structure a set A with binary operation *
commutative let (s,*) be an algebraic structure the operation * is said to be commutative if a*b= b*a for all a,b that element of S
associative let (s,*) be an algebraic structure the operation * is said to be associative if (a*b)*c= a*(b*c) for all a,b,c that element of S
identity element An element e is element of S is said to be identity element for * if a*e =e*a=a for all a in S
inverse If * has an identity element . An element a in S is said to be invertiable if there exists a in S s.t. a* a'=a'*a=e
isomorphism let (s,*) and (T, @) be two algebraic structures a map o: S--->T is called an isomorphism between S and T if o is one to one onto o(a*b)= o(a) @ o(b) for all a,b in S
group a group is a set G , with binary operation * satisfying 3 axioms * is associative there exist an identity element each a in G is invertiable
abelian A group G is abelian if * is commutative
subgroup Let Ge be a group. A non-empty subset h of G is called a subgroup if H itself is a group under the same operation in G
trivial subgroup improper subgroup non- trivial subgroup {e}< G G<G any other subgroup
cyclic subgroup let G be a group and a be in G denoted For n element of natural numbers a^n = a*a*a*.......*a n-copies
order the number of elements of a group denoted |G|
cyclic group let G be a group . we say G is a cyclic group if there is some a in G s.t. <a>={a^n:n in Z}= G
permutation Let A be a non- empty set a permutation on A is a function F: A---> A this one to one and onto
orbit the equivalence class of each a in A under a permutation 6 is called the orbit of a [a]= {b in A: b~a} = {b in A: b =6^n(a) for some n in Z
cycles A permutation 6 in Sn is called cycle if it has at most one orbit that contains more than one element
length the length of the cycle is the number of elements in that orbit containing more than one element
transposition a cycle of length 2
even/ odd permutation A permutation is even or odd depending whether we express it as even number or odd of transposition
coset let H be a subgroup of G, and a in G the set aH={ah: h in H } is call left coset of H in G determined by a similarly Ha= {ha: h in H} is called the right coset of H in G by a.
index the number of distinct left cosets of H in G denoted [G:H] = |G|/|H|
finitely generated a group G is said to be finitely generated if there is a finite set F contained in G s.t F generates G
homorphism let (G,*) and (G', @) be two groups. a map o: G----->G' is called a homorphism if o(a*b)= o(a) @ o(b)
kernel of o let o: G---> G' be a group homomorphism the kernel of o ker(o):= {g in G : o(g)= e'}
normal subgroup A subgroup N of a group G is a normal subgroup if aN= Na for all a in G
G/N {gN:g in G}
factor group let G be a group and N is a normal subgroup of G G/N= {gN:g in G} (aN)(bN)= abN aN=bN cN=dN acN=bdN
automorphism An isomorphism o: G---->G of a group G to itself is called automorphism,
inner automorphism automorphism of the type of ig g in G
simple a group is called simple if G not equal to {e} and it has no non-trivial normal subgroup
ring a ring is a non empty set R with two binary operation + and multi if (R,+) is abelian group multi is associative multiplication is distributive over addition
commutative ring let R be a ring and multiplication is commutative
unit let R be a ring with unity. If a in R is invertible under mulitplication
unity if there is identity element for multiplication it
division ring a ring is called a division ring if it a ring if it a ting with unity and every non zero element is a unit
Created by: jr0211