Hello I am struggling with these 2 questions.

1) Prove that

and

(symmetry group of the equilateral triangle) are isomorphic.

It's quite clear that they are but is there a way to show this without having to build an isomorphism that would take a long time as I would have to show every pair of permutations f,g in

would give

I have this theorem in my notes: Every group G is isomorphic to a subgroup of

for some choice of set X. [X = { bijections f: f: X -> X }]

**In particular, if then G and H are isomorphic for some for some .**
Am I supposed to use this? Or can I just draw a Cayley table of both groups and say look the two groups are algebraically indistinguishable and hence are isomorphic? I found a few proofs on the internet but I don't understand the notation or theorems they use.

2)Let G be a group and suppose that G has subgroups H and K for which the following conditions all hold:

(a) G = HK, (b)

, (c)

, (d)

.

Show also that

.

HINT:

Show first that any element of G can be uniquely expressed in the form hk, where . Then, by considering elements of the form

where

show that every element in H

commutes with every element of K. Finally by considering a suitable mapping from H X K to G obtain the required isomorphism.

OK I can make the isomorphism from H x K to G but in proving it is bijective I need to use commutativity and uniqueness of elements in G so how do i prove these 2 things ( in red ) ? I have pages and pages of h's and k's. I was trying to show that

was in H and in K and therefore in

And of course if

then

Any help and discussion would be appreciated.