Klein 4-group

• Oct 24th 2010, 11:00 AM
mathgirl1188
Klein 4-group
Let H be the Klein 4-group. To which familiar group is the quotient group

Any suggestions or hints would be fabulous!
• Oct 24th 2010, 03:15 PM
HallsofIvy
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 PM
tonio
Quote:

Originally Posted by HallsofIvy
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\$?

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 AM
HallsofIvy
Oops, yes, 24/4= 6, not 8! I really need to work on my arithmetic!