Re: algebra with order 160.

In article <epd52a$qgr$1@xxxxxxxxxxxxxxxx>,
mina_world <mina_world@xxxxxxxxxxx> wrote:

"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
news:epakiu$2qa5$1@xxxxxxxxxxxxxxxxxxxxx
In article <epadu0$t5b$1@xxxxxxxxxxxxxxxx>,
mina_world <mina_world@xxxxxxxxxxx> wrote:
Hello sir~

Prove that no group of order 160 is simple.

------------------------------------------------
I found a solution from sci.math by FinalFntsy..

Let G be a group of order 160 = 5*2^5,
and let P be a Sylow 2-subgroup of G.

[...]

In fact, I want use the normalizer to prove it.
Let's go...

Since |G| = 160 = (2^5)*5,
the number of Sylow 2-subgroups are 1 or 5.

1)
If the number of Sylow 2-subgroup is 1,
Let P be Sylow 2-subgroup.
so, gPg^-1 = P for all g in G by inner automorphism.
so, P is normal.

If there is only one p-Sylow subgroup for a given p, then it must be
normal. This gives you nonsimplicity already.

2)
If the number of Sylow 2-subgroups is 3,

But you already noted that there have to be either 1 or 5, so why are
you dealing with the case where there are 3 of them?

[.snip.]

And if you meant 5, your attack is still likely to be very
complicated.

Since you want to use the normalizer, why not let G act on {S1, S2,
S3, S4, S5}, the set of the five 2-sylow subgroups of G, by
conjugation?

In fact, I am unfamiliar to use the conjugation homomorphism.
Fraleigh book use the normalizer to show in this problems.
I want to know your method.(action + conjugation + non-trivial
homomorphsim)

I know it.
Let X be a G-set.
For each g in G, the function h_g : X -> X defined by
h_g(x) = g.x for x in X is a permutation of X.
Also, the map f : G -> S_X defined by f(g) = h_g is a homomorphism
with the property that f(g)(x) = g.x

How(Where) do you use conjugation ?

Let X = {S1, S2, S3, S4, S5} be the set whose elements are the 2-Sylow
subgroups of G.

Given g in G and x in X, define the action of g on x, g.x, to be

g.x = gxg^{-1}.

That is, g.S1 = g(S1)g^{-1}.

Since the conjugate of a 2-Sylow subgroup is a 2-Sylow subgroup, this
action induces a permutation on X; it is a transitive action, because
given any two elements of X, Si and Sj, ther always exists a g in G
such that g(Si)g^{-i} = Sj, by Sylow's Theorems.

The fact that G acts on X (by conjugation) gives you a homomorphism
from G to S_X, the set of all permutations of X, which of course is
isomorphic to S_5.

--
======================================================================
"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

.

Relevant Pages

  • Re: Group of order 720
    ... N_Pis a subgroup of P; so it must be of order ... The orbit of P' in Syl_3has length ... than one point in the conjugation action on the Sylow 3-subgroups. ... normalizer N of Q contains more than one Sylow 3-subgroup. ...
    (sci.math)
  • Re: Help proving that a simple group of order 360 must be isomorphic to A_6
    ... normalizer of a Sylow 5-subgroup has 10 elements. ... are of prime order, they intersect only trivially. ... a normal Sylow 2-subgroup D. D is also normal in some subgroup of G ...
    (sci.math)
  • Re: algebra with order 160.
    ... I found a solution from sci.math by FinalFntsy.. ... the number of Sylow 2-subgroups are 1 or 5. ... I am unfamiliar to use the conjugation homomorphism. ... Fraleigh book use the normalizer to show in this problems. ...
    (sci.math)
  • Re: Sylow Difficulty
    ... >>to my Sylow problems because I had such difficlty with them. ... Prove that G has a normal subgroup of ... If too many subgroups intersect trivially, ... Then the normalizer of X contains at least 2 ...
    (sci.math)
  • Re: Question on old proof from Derek Holt
    ... so there is a subgroup Q of order 3 (the pointwise stabilizer of this ... subgroup T here is properly contained in a Sylow 2-subgroup S of G, ... So P acts fixed-point-freely on. ... of N. Now S is contained in X_2, where X is the normalizer of N in the ...
    (sci.math)