Hi there vincent!

Swlabr : consider

. Each of

has order

; the other three elements

form a subgroup isomorphic to the cyclic group of three elements. So that's another example.

It is not true that the product of two elements of even order has even order. For example

has order

.

Also you say but there could be an element of order not

which is not of odd order!

Vincent : are you

*given* that the elements of order other than

form a subgroup? Because in general that's not true. For example in

,

has order

(which is not

) but

has order 2.