Let H be the Klein 4-group. To which familiar group is the quotient group

S4/H isomorphic? Justify your answer.

Any suggestions or hints would be fabulous!

Printable View

- Oct 24th 2010, 11:00 AMmathgirl1188Klein 4-group
Let H be the Klein 4-group. To which familiar group is the quotient group

S4/H isomorphic? Justify your answer.

Any suggestions or hints would be fabulous! - Oct 24th 2010, 03:15 PMHallsofIvy
My only suggestion would be to sit down and

**do**it! $\displaystyle S_4$, the group of permutations on 4 "symbols", contains 4!= 24 elements. It is tedious but not impossible to write out its operation table. It has a subgroup isomorphic to the Klein group (if it didn't this problem would make no sense). Can you identify that subgroup? $\displaystyle S_4/H$ has 8 members. Can you work out the operation table for it from the operation table for $\displaystyle S_4$? - Oct 24th 2010, 08:57 PMtonio

A little typo slip there: $\displaystyle S_4/H$ has 6 elements, of course, and since there are only two different groups,

up to isomorphism, of order 6, there are no many options from where to choose...

Now, remember that a quotient group is abelian iff the normal subgroup contains the derived, or commutator, subgroup

of the whole group, which is here $\displaystyle A_4$ , so...

Tonio - Oct 25th 2010, 05:12 AMHallsofIvy
Oops, yes, 24/4= 6, not 8! I really need to work on my arithmetic!