| Term | Definition |
|---|
| function composition | a binary operation on the collectin of all permutations of a set A |
| function composition operation | permutation multiplication |
| S(mini A) | collection of all permutations of A (a nonempty set): under permutation multiplication: closed under operation: associative |
| S(mini n) | group of all permutations of A (finite set {1,2,...n})=symmetric group on n letters |
| # of elements in S(mini n) | n(factorial) |
| abelian | a group that is commutative; any group of at most 4 elements |
| D(mini 3) | the third dihedral group; symmetries of an equilateral triangle |
| D(mini n) | the nth dihedral group; the group of symmetries of the regular n-gon |
| octic group | the group of symmetries of the square (D(mini 4)): non-abelian |
| Cayley's Theorem | every group is isomorphic to some group consisting of permutations under permutation multiplication |
| equivalence class in A | determined by the equivalence relation are the orbits of sigma |
| cycles are disjoint iff | any integer is moved by at most one of these cycles; thus no # appears in the notations of two different cycles |
| multiplication of disjoint cycles | is commutative |
| transposition | a cycle of length 2; leaves all elements but two fixed & maps each of these onto the other |
| cycle | a product of transpositions (any permutation of a finite set of at least two elements) |
| even permutation | a finite set that can be expressed as a product of an even number of transpositions |
| odd permutation | a finite set that can be expressed as a product of an odd number of transpositions |
| alternating group A(mini n) on n letters | the subgroup of S(mini n) consisting of the even permutations of n letters |
| left coset of H in G | (given H<G and g exists in G) represented by g: gH={gh such that h exisits in H} (note this is a subset of G) |
| right coset of H in G | (given H<G and g exists in G) represented by g: Hg={hg such that h exists in H} (note this is a subset of G) |
| normal subsets | when right coset=left coset |
| H=<[3]> | H is the cyclic subgroup generated by 3 |
| aH=bH iff | a(inverse)b exists in H |
| Ha=Hb iff | ba(inverse) exists in H |
| Lagrange's Theorem | Let G be a finite group (no definition of commutativity), Let H<G, Then the order of H is a divisor of the order of G |
| factor group of G by H | suppose G is a group, H<G, and left coset=right coset, then calling this coset set G/H we have (G/H, *) is a group |
| G1XG2 is abelian iff | both G1 and G2 are abelian |
| index (G:H) of H in G | the number of left cosets of H in G (given that H is a subgroup of a group G) |
Combine with other sets
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