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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
125
2011
0
On the Rarity of Quasinormal Subgroups
John
Cossey
Australian National University, CANBERRA, AUSTRALIA
Stewart
Stonehewer
University of Warwick, COVENTRY, UNITED KINGDOM
For each prime $p$ and positive integer $n$, Berger and Gross have defined a finite $p$-group $G=HX$, where $H$ is a core-free quasinormal subgroup of exponent $p^{n-1}$ and $X$ is a cyclic subgroup of order $p^n$. These groups are universal in the sense that any other finite $p$-group, with a similar factorisation into subgroups with the same properties, embeds in $G$. In our search for quasinormal subgroups of finite $p$-groups, we have discovered that these groups $G$ have remarkably few of them. Indeed when $p$ is odd, those lying in $H$ can have exponent only $p$, $p^{n-2}$ or $p^{n-1}$. Those of exponent $p$ are nested and they all lie in each of those of exponent $p^{n-2}$ and $p^{n-1}$.
General
81
105
10.4171/RSMUP/125-6
http://www.ems-ph.org/doi/10.4171/RSMUP/125-6