They may not be the same but they are similar.
Originally Posted by Moo
Proof is simple: Suppose (and therefore If then Therefore by contrapositivity,
In bmp05’s original example, is the addibive group of the reals and is the additive group of the rationals; hence rational + irrational = irrational. In the odd/even example, is the additive group of the integers and is the subgroup of the even integers – even + odd = odd.
(i) Even permutation × odd permutation = odd permutation. (Take to be a symmetric group, to be the corresponding alternating group.)
(ii) Rotation × reflection = reflection. (Take to be a dihedral group, to be the subgroup of rotations.)