Help
Options
Didn't know it?
click below
click below
Knew it?
click below
click below
Don't Know
Remaining cards (0)
Know
retry
shuffle
restart
0:00
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.
Normal Size Small Size show me how
Normal Size Small Size show me how
Abstract Definitions
| Term | Definition |
|---|---|
| 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
Popular Math sets