From: lounesto@dopey.hut.fi (Pertti Lounesto)
Newsgroups: sci.math
Subject: Q: Extension of automorphism of a subgroup
Date: 20 Jan 1995 15:07:04 GMT
Let G be a group, and H its subgroup, and a:H->H an automorphism of H.
Is there always an automorphism of G, whose restriction to H is a, and
if not, could you give a counter-example for the finite group (of lowest
order)?
--
Pertti Lounesto E-Mail: lounesto@dopey.hut.fi
Helsinki Univ. of Technology
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From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Q: Extension of automorphism of a subgroup
Date: 20 Jan 1995 16:47:35 GMT
In article ,
Pertti Lounesto wrote:
>Let G be a group, and H its subgroup, and a:H->H an automorphism of H.
>Is there always an automorphism of G, whose restriction to H is a, and
>if not, could you give a counter-example for the finite group (of lowest
>order)?
Let G be the dihedral group of order 8 (the group of symmetries of
the square). It has an elementary-abelian subgroup H of order 4 generated
by the reflections across the edges of the square. Now, H itself has
6 automorphisms (arbitrarily permute the 3 elements of order 2). But
not all of these extend to the whole group G: H includes the 180-degree
rotation of the square, which is in the center of G; the other
involutions of H are not in the center of G, so no automorphism of G
can permute these.
dave
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From: lounesto@dopey.hut.fi (Pertti Lounesto)
Newsgroups: sci.math
Subject: Re: Q: Extension of automorphism of a subgroup
Date: 21 Jan 1995 14:24:57 GMT
On 20 Jan 1995 rusin@washington.math.niu.edu (Dave Rusin) gave a nice
answer to my question, thank you. Your response in fact resolved a
problem I had for a while, and your counter-example works as such also
in my much more abstract case. Here was my problem: Does the triality
automorphism of Spin(8) extend to Pin(8)? Triality permutes the
three non-trivial elements {-1, e12...8, -e12...8} in the center of
Spin(8), but only -1 is in the center of Pin(8). So your counter-
example shows that triality does not extend to Pin(8).
Dave Rusin, thank you once more for your nice resolution.
--
Pertti Lounesto E-Mail: lounesto@dopey.hut.fi
Helsinki Univ. of Technology
Newsfeed unreliable. I might have missed the beginning of this thread, so
forgive me if I'm repeating facts in posts that have expired on my system.
If you post a follow-up to this article, send it to me by E-mail, as well.
==============================================================================
Newsgroups: sci.math
From: william@math.Princeton.EDU (William Schneeberger)
Subject: Re: Q: Extension of automorphism of a subgroup
Date: Sat, 21 Jan 1995 21:51:21 GMT
In article lounesto@dopey.hut.fi (Pertti Lounesto) writes:
>Let G be a group, and H its subgroup, and a:H->H an automorphism of H.
>Is there always an automorphism of G, whose restriction to H is a, and
>if not, could you give a counter-example for the finite group (of lowest
>order)?
>--
> Pertti Lounesto E-Mail: lounesto@dopey.hut.fi
> Helsinki Univ. of Technology
>
>Newsfeed unreliable. I might have missed the beginning of this thread, so
>forgive me if I'm repeating facts in posts that have expired on my system.
>If you post a follow-up to this article, send it to me by E-mail, as well.
Someone else pointed out that D4 would be the G of minimal order, with
H of order 4. However, we can get H to be of order 3.
In the nonabelian group of order 21, the semidirect product of C7 with
C3, we can find an element a of order 3 such that for any x of order
7, x^a=x^2, but x^(a^2)=x^4. So the group admits an automorphism
which does not extend to the larger group.
--
Will Schneeberger Hi There !!
william@math.Princeton.EDU
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From: LAURAHELEN@news.delphi.com (LAURAHELEN@DELPHI.COM)
Newsgroups: sci.math
Subject: Re: Q: Extension of automorphism of a subgroup
Date: 21 Jan 1995 19:04:09 -0500
>In general a counterexample is a direct sum of abelian groups with the same
>order. Any permutation of the generators induces an automorphism of the
>direct sum. Now embed this direct sum into a direct sum where one factor
>is a cyclic group of higher order. Any permutation of the original
>generators which involves the factor that was extended , can't be lifted
>to the larger direct sum.
I meant "direct sum of cyclic groups with the same order".
Anyway, this style of counterexample does not seem to work in the case
where G is finite order, H < G, and |H| and |G|/|H| are relatively prime.
So I wonder whether an automorphism of H can always be lifted to G in
this case.
*she goes off to mull*
Laura
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From: ellenber@husc7.harvard.edu (Jordan Ellenberg)
Newsgroups: sci.math
Subject: Re: Q: Extension of automorphism of a subgroup
Date: 21 Jan 95 22:06:12
LAURAHELEN asked whether, if G is a finite group, H a subgroup of G with
|H| and [G:H] relatively prime, an automorphism of H can be lifted to
an automorphism of G.
I think the answer is still no. Let H be the direct sum of four groups of
order 2 and let h1,h2,h3,h4 be the generators of H. Let G be the group
of order 48 generated by H and g, where g^3=1 and conjugation by g
permutes h2,h3,h4 cyclically. Then G and H are of the desired form, but
the automorphism of H transposing h1 and h2 does not lift to G, since h1 is
central in G and h2 is not.
Jordan Ellenberg
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From: LAURAHELEN@news.delphi.com (LAURAHELEN@DELPHI.COM)
Newsgroups: sci.math
Subject: Re: Q: Extension of automorphism of a subgroup
Date: 21 Jan 1995 22:40:16 -0500
LAURAHELEN@news.delphi.com (LAURAHELEN@DELPHI.COM) writes:
>Anyway, this style of counterexample does not seem to work in the case
>where G is finite order, H < G, and |H| and |G|/|H| are relatively prime.
>So I wonder whether an automorphism of H can always be lifted to G in
>this case.
It can't always. Let H = C_3 x C_3, generated by b and c. Let a be an
element of order 2. Let G be the semidirect product of with H; let
a act on H by f(a)(b) = b^2, f(a)(c) = c. Then there is an automorphism
s of H which exchanges b and c, but it can't be lifted to G because c is in
the center of G, while b is not.
But if G is abelian then G is the direct product G/H x H and the automorphism
can be lifted.
Laura