Re: Automorphism group & Centre

In article <45ef06ba$0$17944$4fafbaef@xxxxxxxxxxxxxxxxxxx>,
bluelabel <bluelabel.invalid@xxxxxxxx> wrote:

"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote:
bluelabel <bluelabel.invalid@xxxxxxxx> wrote:
Let K_{j-1} <] K_j <] K_{j+1} be three sugroups of a group G.
K_{j+1}/K_j is cyclic of order p and K_j/K_{j-1} is cyclic of order q
(where
p<q, p and q are primes).

I read that:
since |Aut(K_j/K_{j-1})|= q-1, which is not divisible by p,

That q-1 is not divisible by p does not follow from the assumptions
you have given so far, and must be explicitly stated in
advance. Otherwise you could have p=2 and q an odd prime, or something
like p=3 and q=7. So you are ->also<- assuming that p does not divide
q-1.



Yes, it escaped me. I had to write "p and q are ODD primes" instead of "p
and q are primes".

That is not enough, as I noted above: if p=3 and q=7, then p and q
are both odd primes, distinct, but p=3 divides q-1=6. You need to
EXPLICITLY say that p does not divide q-1.

[...]

%-)
I admit that I would never have solved it myself in a century. There is a
passage that is not clear to me.
Let me rename K_{j-1} as "M".
Well, if I understood well, you said that K_{j+1}/M acts on K_{j}/M:

gM |-----> gM.yM.(gM)^{-1} = gyg^{-1}M .

with gyg^{-1}M in K_{j}/M (this is due to the fact that K_j is normal in
K_{j+1} ).
So it is induced a morphism

F: K_{j+1}/M ---> Aut(K_j/M)
gM |---> f

such that f : yM |--> gyg^{-1}M.
After that, you said that every element of order p should have been mapped
trivially. I take this as a consequence of the homomorphism properties (for
example, an element of order p must be mapped to an element whose order
divides p). Is it so or is there a more complex reason behind?

No, just that (and the fact that you know Aut(K_j/M) has order that is
relatively prime to p).

--
======================================================================
"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: Automorphism group & Centre
    ... That q-1 is not divisible by p does not follow from the assumptions ... I had to write "p and q are ODD primes" instead of "p ... EXPLICITLY say that p does not divide q-1. ...
    (sci.math)
  • Re: Automorphism group & Centre
    ... bluelabel wrote: ... p and q are primes). ... That q-1 is not divisible by p does not follow from the assumptions ... that is, x commutes with all g, hence is central. ...
    (sci.math)
  • Re: Automorphism group & Centre
    ... p and q are primes). ... That q-1 is not divisible by p does not follow from the assumptions ... are both odd primes, distinct, but p=3 divides q-1=6. ... EXPLICITLY say that p does not divide q-1. ...
    (sci.math)
  • Re: Automorphism group & Centre
    ... That q-1 is not divisible by p does not follow from the assumptions ... I had to write "p and q are ODD primes" instead of "p ... EXPLICITLY say that p does not divide q-1. ...
    (sci.math)
  • Re: Automorphism group & Centre
    ... That q-1 is not divisible by p does not follow from the assumptions ... Otherwise you could have p=2 and q an odd prime, ... I had to write "p and q are ODD primes" instead of "p ... So it is induced a morphism ...
    (sci.math)