Finite Groups, A 5 cycle and a Double Transposition
A 5 cycle and a Double Transposition
Let G be an even permutation group on 5 letters,
containing a 5 cycle and an involution.
The involution must be two transpositions running in parallel.
Label the elements so that the 5 cycle is a circular shift
of the digits 0 through 4,
and the involution fixes 0.
Consider the 3 ways we can doubly transpose 1 through 4.
If we swap 1 and 2, and 3 and 4,
follow this with a right shift to produce a 3 cycle.
This combines with the 5 cycle to
build
A5.
If 1 and 3 swap, follow this up with a double shift to the right, giving another 3 cycle.
Finally, 1234 could reverse, i.e. reflected through a mirror.
This gives
D5, the
dihedral group of order 5.