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A permutation is a cycle or product of disjoint cycles

Permutation group

cycle notation, order of cycle

If d is positive divisor of n then number of elements of order d in (a) is phi(d)group theory gallia

If d is positive divisor of n then number of elements of order d in (a) is phi(d)

for each positive divisor k of n, the group (a) has exactly one subgroup of order k

If order of a cyclic group is n, then order of every subgroup is divisor of n

Every subgroup of a cyclic group is cyclic, fundamental th. of algebra

Fundamental theorem of algebra

corollary: Let G be a group and a belongs to G with o(a)=n and if a*k=e then n/k

Let G be cyclic group of order n then a_k is generator of G if gcd(k, n)=1

let G be a group and a belong to a to G has infinite order, then all distinct power of the a are

symmetric group of squares

cyclic group, order of group and example

General Linear group of invertible matrices, GL(2, Z7) is a group,

Let H be nonempty finite subgroup of group G then H is subgroup if it is closed

Let G be an ablian group then H containing elements of whose square is identity is subgroup

2Z is subgroup of Z under addition

example of order of group and elements of group

Inverse of a matrix in GL(2,Z11)

definition of congruence modulo and example

Order of group and elements

In a group inverse of ab=inverse of b inverse of a

inverse of inverse a= a