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# 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