Prove/disProve : if G is isomorphic to a subgroup of H & H
is isomorphic to a subgroup of G,then G is isomorphic to H
2006-06-30 22:23:54 · 3 answers · asked by kukur_diamond 1 in Science & Mathematics ➔ Mathematics
This is true in the case of finite groups:
Recall part of what it means to be an isomorphism.
The function is 1 to 1 and onto.
So let g be an isomorphism from G to a subrgroup of H and h be an isomorphism from H to a subgroup of G.
then since g is an isomorphism, |G| <= |H| and since h is an isomorphism, |H| <= |G|, so |G| = |H| and thus g must actually be from G to all of H and h must be from H to all of G. so H and G are isomorphic.
If you're not dealing with finite groups, I'm not sure how to proceed on this.
2006-07-03 15:17:38 · answer #1 · answered by fatal_flaw_death 3 · 0⤊ 0⤋
Not so! Let G be a countable product of Q's under addition, and H=G direct product with Z under addition. Every element in G is divisible by 2, but not so for H.
2006-07-03 18:12:11 · answer #2 · answered by Steven S 3 · 0⤊ 0⤋
property of exhibiting different structure but having the same molecular formula.very common in organic compounds.
2006-06-30 22:29:06 · answer #3 · answered by raj 7 · 0⤊ 0⤋