Re: Help understanding an argument of Phillip Hall


In article <de2q2t$jl7$1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
<mareg@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

>In article <de1thj$euo$1@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin) writes:
>
>I have not looked at the paper itself, but let me nevertheless make a few
>comments.
>
>>I'm hoping someone can help me figure out an argument of Phillip Hall's,
>>which appears in his paper "The Classification of prime-power groups",
>>J. fur Reine Angew. 182 (1940), pp. 130-141.

[...]

>>The maximal stem group is what I would call the nilpotent product of four
>>cyclic groups of order p, or alternatively the relatively free group of
>>rank 4 in the variety of groups of class 2 and exponent p.
>
>Surely, the maximal stem group is not unique?

Yes, sorry. It is the maximal family which is unique, and any two
maximal stem groups will have the same order and be isoclinic, as
usual. The family depends only on the choice of central quotient. In
the specific case at hand the maximal family has rank 10, meaning the
stem groups are of order p^{10}.

>There are others, which do
>not have exponent p. When p is odd, the one of exponent p is somehow
>the natural one to take, but when p = 2, there is no natural choice.
>But whichever maximal stem group you choose, it is true that every group
>whose central quotient is isomorphic to K will be isoclinic to a quotient
>of it.

Yes, you are correct. Thank you for the correction and sorry for any
confusion.

>>This group has
>>a commutator subgroup which is elementary abelian of order p^6, freely
>>generated by the commutators [a_j, a_i], 1 <= i < j <= 4. So the group
>>is of order p^{10}. Any group of order p^5 with central quotient
>>isomorphic to K will be the result of moding out by a subgroup of order
>>p^5 of this commutator subgroup;
>
>This does not seem quite right, because not every such group has exponent p.

Again, my mistake. I've been working with groups of exponent p, so I
mentally made the restriction without realizing.

--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.