Re: Group Theory Question

In article <1177616995.707751.201250@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Craig Feinstein <cafeinst@xxxxxxx> wrote:
On Apr 26, 12:49 pm, Craig Feinstein <cafei...@xxxxxxx> wrote:
First of all, for everyone's amusement, watch this:http://www.youtube.com/watch?v=UTby_e4-Rhg

Secondly, consider the simple group G of order 168 of the
automorphisms of the Fano plane talked about here:http://en.wikipedia.org/wiki/Fano_plane

Such a group is a subgroup of S_7, the permutation group of
{1,2,3,4,5,6,7}.

My question is what is the structure of the group G/S_7, of order 30?
I know that there are four possibilities:

C_30
D_15
D_5 x C_3
D_3 x C_5

(C means cyclic and D means dihedral. Seehttp://www.math.usf.edu/~eclark/algctlg/small_groups.html)

Which one of these possibilities describes G/S_7 and why?

Craig

I'm sorry. I meant S_7/G.

S_7 has only three normal subgbroups, none of which of index 30. So
S_7/G cannot mean a quotient group. And if it means simply the left
cosets of G as a subgroup of S_7, then it is not a group (since G is
not normal in S_7), so the answer to "what is the structure of the
group S_7/G?" is: "it is not a group."

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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