# [GAP Forum] groups of order p^n and exponent p

Alan Camina A.Camina at uea.ac.uk
Fri Dec 3 13:08:18 GMT 2004

```I think that Graham Higman's lower bounf was done by looking at class 2
groups of exponent at most p^2.

Alan

School of Mathematics
University of East Anglia
Norwich
UK NR4 7TJ

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On Fri, 3 Dec 2004, Siddhartha Sarkar wrote:

>
> Dear Prof. Stefan Kohl,
>
>          The result you mentioned about the number of isomorphism types
> of groups of order p^n, is probably by C.Sims London Math.
> Soc.(3)15(1965) 151-166. Is there any similar result if one fixes the
> exponent. Specially, is it known in case exponent p? Kindly reply.
>
>          Thanks to both of you and Prof. Bettina Eick for the response
> of my earlier mail(s); it was really of help. Sorry for including a long
> mail last time. It was a mistake.
>
> regards,
> Siddhartha Sarkar
>
>> ------------------------------
>>
>> Message: 5
>> Date: Wed, 24 Nov 2004 17:59:40 +0100
>> From: Stefan Kohl <kohl at mathematik.uni-stuttgart.de>
>> Subject: Re: [GAP Forum] commutator relations in p-group
>> To: GAP Forum <forum at gap-system.org>
>> Message-ID: <41A4BDFC.6060308 at mathematik.uni-stuttgart.de>
>> Content-Type: text/plain; charset=us-ascii; format=flowed
>>
>> Dear Forum,
>>
>> Siddhartha Sarkar wrote:
>>
>>> I need to know the classification of p-groups satisfy the following
>>> commutator relation:
>>>
>>> (1) say the group G is minimally generated by d elements
>>>                                               x_1, x_2,...,x_d.
>>> (2) if d is even, the relation is [x_1, x_2]...[x_{d-1},x_d] = 1
>>> (3) for d odd, the relation is [x_1, x_2]...[x_d,x] = 1
>>>      for some element x in G.
>>
>> Your condition is relatively weak --
>>
>> A pc-presentation of a p-group of order p^n requires n(n+1)/2 relations.
>> Out of these you prescribe only one.
>>
>> Given that there are asymptotically p^((2/27)n^3) isomorphism types
>> of groups of order p^n, it seems likely that there also is a huge
>> number of p-groups which satisfy your relation, and that it does not
>> make sense to attempt to classify them in general.
>>
>> Bettina Eick has already suggested you off-list to use NQ or ANUPQ --
>>
>> By the way: when replying to Forum digests, please make sure that
>> you only cite the relevant part. Thanks!
>>
>> Best wishes,
>>
>>      Stefan Kohl
>>
>
>
>
>
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