Re: groups
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Sun, 16 Mar 2008 22:20:57 +0000 (UTC)
In article <32f8fdc6-3c0a-49f1-8d21-c9dc27fc3a95@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<judysisley@xxxxxxxxx> wrote:
Can somebody describe the structure of C(A_4) and C(D_6) group of
order 12?
Yes.
You probably can too, if you think about it. You are looking for the
centers. What permutations of 4 letters commute with EVERY
permutation of 4 letters? You can use the fact that they must commute
with each transposition, and that (a,b)sigma = sigma(a,b) is the same
as (a,b)sigma(a,b) = sigma. This in turn means that if you write sigma
as a product of disjoint cycles, and then you replace every a by b and
every b by a in that expression, you will just get another way to
write sigma. From there, it should be easy.
As for the center of the dihedral group of 12 elements, an element
r^i*s^j, with 0 <= i <= 5 and 0 <= j <= 1 is in the center if and only
if it commutes with both r and s (r represents the rotation, s the
reflection). Since sr = r^{-1}s, when j=0 consider what happens when
you take sr^i and r^is; and when j=1 consider what happens when you
take r(r^is) and (r^is)r.
--
======================================================================
"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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