The center of a Shunkov group with one saturation condition
Āl. Shlyopkin, I. Sabodakh
A group \(G\) is saturated with groups in a set \(X\) if every finite
subgroup \(K\) of \(G\) is contained in a subgroup of \(G\)
isomorphic to some group in \(X\), see [1].
Let \(K\) a the finite subgroup of \(G\) that is saturated with a set \(X\).
The set of all subgroups of \(G\) that contain \(K\) and
are isomorphic to groups in \(X\) will be denoted by \(X(K)\). In
particular, if \(1\) is the identity subgroup of \(G\) then \(X(1)\) is the set of all
subgroups of \(G\) isomorphic to groups in \(X\), see [2].
Let \(\Im=\{GL_{2}(p^{n})\}\), where \(p\) is a fixed prime and
\(n\) is not fixed.
We continue the research started in [3, 4, 5]. The following result has been proved:
Theorem 1.Let \(G\) be a periodic Shunkov
group saturated with a set \(\Im\) and let \(K\in\Im(1)\). Then
\(Z(K)\subset Z(G)\) and \(Z(G)\) is a locally cyclic group.
References
A. K. Shlyopkin, Conjugate biprimitive finite groups that
contain no solvable subgroups, 3rd Internat. Alg Conf., Abst., Krasnoyarsk (1993), 363. (Russian)
A. A. Kuznetsov, K. A. Philippov, Groups saturated with a given
set of groups, Sib. Elect. Math. Reports, 8 (2011), 230–246. (Russian)
D. N. Panushkin, Shunkov groups saturated with direct products
of different groups, Ph.D. Phys.-Math. Thesis, Krasnoyarsk (2010), 66. (Russian)
A. A. Shlyopkin, Periodic groups saturated by the groups \(GL_2(p^n)\),
Internat. Alg. Conf, Abst., Kyiv (2012), 144.
A. A. Shlyopkin, On groups saturated with \(GL_2(p^n)\), Vestnik SibGAU,
1 (2013), 100–108. (Russian)
