Congruence condition fails for abelian subgroups of prime-sixth order
From Groupprops
Statement
Let be any prime number. Then, there exists a finite -group of order that contains exactly two abelian subgroups of order , both of which are elementary abelian normal subgroups. Thus:
- The collection of abelian groups of order fails to satisfy a universal congruence condition for any prime .
- The singleton collection of the elementary abelian group of order fails to satisfy a universal congruence condition for any prime .
Related facts
See also collection of groups satisfying a universal congruence condition#Examples/facts.
- Abelian-to-normal replacement fails for prime-sixth order for prime equal to two
- Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one: This in particular shows that for , the existence of an abelian subgroup of order implies the existence of an abelian normal subgroup of order .
- Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime: For odd, the universal congruence condition does hold for abelian subgroups of order , for any fixed between and .
Proof=
Further information: free product of class two of two elementary abelian groups of prime-square order
Let be the quotient of the free product of two elementary abelian groups of order by the third member of its derived series. Thus, is the free product of class two of two elementary abelian groups of order . Then, is a group of order . We claim that has exactly two abelian subgroups of order , and both of these are elementary abelian and normal.
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Journal references
- Abelian subgroups of p-groups, an algebraic approach by David Jonah and Marc Konvisser, Journal of Algebra, ISSN 00218693, Volume 34, Page 386 - 402(Year 1975): ^{Official copy}^{More info}: This describes the construction of counterexamples.
- Counting abelian subgroups of p-groups: a projective approach by Marc Konvisser and David Jonah, Journal of Algebra, ISSN 00218693, Volume 34, Page 309 - 330(Year 1975): ^{PDF (ScienceDirect)}^{More info}: This is a companion paper that proves the opposite results for orders up to .
- Large abelian subgroups of p-groups by Jonathan Lazare Alperin, Transactions of the American Mathematical Society, Volume 117, Page 10 - 20(Year 1965): ^{Official copy}^{More info}: A paper published in 1965 that explores similar questions.