Let \(\pi\) be a set of primes. A periodic group \(G\) is called a \(\pi\)-group if all of its element orders are divisible only by the primes in \(\pi\). A group \(G\) acts freely on a nontrivial group \(V\), and the action is called a free action, if \(vg=v\) implies \(v=1\) or \(g=1\) for \(v\in V\) and \(g\in G\).
Our goal is to describe \(\{2,3\}\)-groups acting freely on Abelian groups.
The interest in this topic was raised by the work of E. Jabara and P. Mayr [1]
who proved that a \(\{ 2,3\}\)-group \(G\) of finite period
\(2^m3^2\) that acts freely on an Abelian group is locally finite. Later, D. V. Lytkina showed that this result is also true when we do not assume that the period of a Sylow 2-subgroup of \(G\) is finite.
A locally cyclic group is a group whose every finite subset is contained in a cyclic subgroup.
A quaternion group is a quaternion group of order \(8\) or a generalized quaternion group, i. e. a group isomorphic to
\[
Q_{2^{m+1}}=\langle a,b\ |\ a^{2^m}=b^4=1, \ a^b=a^{-1},\ b^2=a^{2^{m-1}}\rangle,\quad m\geqslant 3.
\]
A locally quaternion group is a 2-group whose every finite subset is contained in a quaternion subgroup.
The quaternion group of order \(8\) possesses an automorphism of order 3.
The corresponding semidirect product of order 24 is isomorphic to the group \(SL_2(3)\).
Denote by \(\tilde S_4\) an extension of a group of order 2 with a group \(S_4\) of degree 4, which has a quaternion Sylow 2-subgroup.
A group \(G\) is called a central product of its subgroups \(A\) and \(B\) with union by a subgroup \(C\), if \(G=AB\), \([A,B]=1\) and \(C=A\cap B\). The central product \(AB\) is said to be nontrivial, iff \(A\ne G\ne B\).
Let \(p\) be an odd prime and \(a\) a positive integer. We say that an infinite \(p\)-group \(P\) is a group of type \(Q(p^a)\) (or a \(Q(p^a)\)-type group), if \(P\) possesses the following properties:
\(\quad 1.\ \) \(Z(P)\) is a cyclic group of order \(p^a\).
\(\quad 2.\ \) Every finite subgroup of \(P\) is cyclic.
Note that groups of type \(Q(p^a)\) are not locally finite.
We say that a \(\{ 2,p\}\)-group \(H\) with an involution is a group of type \(Q(p^a,d)\) if \(H\) possesses the following properties:
\(\quad 1.\ \) \(Z(H)\) is locally cyclic.
\(\quad 2.\ \) Every 2-element of \(H\) belongs to \(Z(H)\); all 2-elements form a group \(T\) of order \(2^d\) (it is possible that \(d=\infty\)).
\(\quad 3.\ \) \(H/T\) is a group of type \(Q(p^a)\) for some positive integer \(a\).
Theorem 1.
Let \(G\) be a \(\{ 2,3\}\)-group that acts freely on an Abelian group. Then one of the following statements is true.
\(\ \ 1)\ \) \(G\) is locally finite and isomorphic to one of the following groups:
\(\qquad \bullet\ \) a locally cyclic group;
\(\qquad \bullet\ \) the direct product of a locally cyclic \(3\)-group with a locally quaternion group;
\(\qquad \bullet\ \) the semidirect product of a locally cyclic \(3\)-group \(R\) with a cyclic \(2\)-group \(\langle b\rangle\), where \(b^2\ne 1\)
and \(a^b=a^{-1}\) for every \(a\in R\);
\(\qquad \bullet\ \) a semidirect product of a locally cyclic \(3\)-group \(R\) with a locally quaternion group \(Q\), where \(|Q:C_Q(R)|=2\);
\(\qquad \bullet\ \) the semidirect product of a quaternion group \(Q_8=\langle x,y\rangle\)
of order \(8\) with a cyclic \(3\)-group \(\langle a\rangle\), where \(x^a=y\);
\(\qquad \bullet\ \) the group \(\tilde S_4\).
\(\ \ 2)\ \) \(G\) is not locally finite and all prime-order elements of \(G\) generate a cyclic subgroup.
Any of the groups mentioned can act freely on some Abelian group.
In Theorem 2, we fully describe the groups from item 2) of Theorem 1.
Theorem 2.
Let \(G\) be a non-locally-finite \(\{ 2,p\}\)-group,
where \(p\) is an odd prime. All prime-order elements of \(G\) generate a cyclic subgroup iff all \(2\)-elements of \(G\) generate a \(2\)-subgroup \(S\), which is locally cyclic or locally quaternion. Besides, one of the following conditions is true:
\(\ \ 1)\ \) \(G=P\times S\), where \(P\) is a \(Q(p^a)\)-type group;
\(\ \ 2)\ \) \(S\) is a nontrivial locally cyclic group, and \(G\) is of \(Q(p^a,d)\)-type;
\(\ \ 3)\ \) \(S\) is a locally cyclic group, and \(G\) is a non-trivial central product of \(S\) and a \(Q(p^a,d)\)-type group with union by a subgroup of order \(2^d\);
\(\ \ 4)\ \) \(S\) is a locally quaternion group, and
\(G\) is a central product of \(S\) and a \(Q(p^a,1)\)-type group with union by a subgroup of order \(2\);
\(\ \ 5)\ \) \(S=Q_8\), \(p=3\), \(|G:C_G(S)|=3\), and \(G/S\) is a \(Q(p^a)\)-type group.
Here \(C_G(S)\) is either a \(Q(p^a,1)\)-group, or a central product of a \(Q(p^a)\)-type group and a group of order \(2\).
Acknowledgement.The work is
supported by Russian Foundation of Basic Research
(Grants 12-01-90006, 13-01-00505,
11-01-00456), the Federal Target Program
(Contract No. 14.740.11.0346), and the Integration Project of the
Siberian Division of the Russian Academy of Sciences for
2012-2014 (No. 14).
References
E. Jabara, P. Mayr, Frobenius complements of exponent dividing \(2^m\cdot 9\), Forum Math.,
21, N1 (2009), 217–220.
D. V. Lytkina, Periodic groups acting freely on abelian groups, Algebra and Logic,
49, N3 (2010), 256–264.
