# Subgroups generated by transpositions #

This file studies subgroups generated by transpositions.

## Main results #

`swap_mem_closure_isSwap`

: If a subgroup is generated by transpositions, then a transposition`swap x y`

lies in the subgroup if and only if`x`

lies in the same orbit as`y`

.`mem_closure_isSwap`

: If a subgroup is generated by transpositions, then a permutation`f`

lies in the subgroup if and only if`f`

has finite support and`f x`

always lies in the same orbit as`x`

.

If the support of each element in a generating set of a permutation group is finite, then the support of every element in the group is finite.

Given a symmetric generating set of a permutation group, if T is a nonempty proper subset of an orbit, then there exists a generator that sends some element of T into the complement of T.

If a subgroup is generated by transpositions, then a transposition `swap x y`

lies in the
subgroup if and only if `x`

lies in the same orbit as `y`

.

If a subgroup is generated by transpositions, then a permutation `f`

lies in the subgroup if
and only if `f`

has finite support and `f x`

always lies in the same orbit as `x`

.

A permutation is a product of transpositions if and only if it has finite support.

A transitive permutation group generated by transpositions must be the whole symmetric group

A transitive permutation group generated by transpositions must be the whole symmetric group