- #26

Deveno

Science Advisor

- 906

- 6

I've just thinking about the previous part.

[itex]S_3 \times S_3 = \{ e, (12), (23), (13), (123), (132) \} \times \{ e, (45), (56), (46), (456), (465) \}[/itex]

isnota copy of [itex]S_3 \times S_3[/itex] inside [itex]S_6[/itex] because it is not a subgroup of [itex]S_6[/itex] (subgroups of [itex]S_6[/itex] consist of elements of [itex]S_6[/itex]).

How can I convert thepairsof elements into elements of [itex]S_6[/itex]?

that's a good point.

this is a specific example of a more general theorem:

suppose A,B are subgroups of a finite group G such that:

ab = ba, for all a in A and b in B, and that:

A∩B = {e}.

prove that:

a) AB = {ab: a in A, b in B} is a subgroup of G

b) AxB is isomorphic to AB

note that since G is finite, you only need to show closure to prove (a). to prove (b), it suffices to show that if ab = e, then a = b = e (why?).