Abstract
Full Text
UDC 519.44
MATHEMATICS
M. M. GOL'DENBERG, N. F. SESEKIN
ON GROUPS WITH INVARIANT NOT QUITE SPLITTABLE SUBGROUPS
(Presented by Academician V. M. Glushkov, 7 I 1970)
- A subgroup (H) of a group (G) is called a (\theta)-subgroup if it has property (\theta), and a (\bar{\theta})-subgroup in the opposite case. A number of works of recent years have been devoted to the study of groups in which every (\bar{\theta})-subgroup is invariant (we shall call such groups (\bar{\theta}J)-groups; see the literature in the survey of S. N. Chernikov ((^1))), where cyclicity, commutativity, nilpotency, and certain other properties were taken as (\theta). It is convenient to include among the (\bar{\theta}J)-groups Dedekind groups and, on the other hand, groups in which every proper subgroup is a (\theta)-subgroup. If (\theta) is an absolute group-theoretic property inherited by subgroups of (\theta)-groups, but not conversely, then knowledge of (\bar{\theta})-groups, all of whose proper subgroups are (\theta)-subgroups, plays an essential role in describing (\bar{\theta}J)-groups. Indeed, in these cases the factor group of a (\bar{\theta}J)-group by its own minimal (\bar{\theta})-subgroup is Dedekind, so that almost the entire commutant of the group under study turns out to be known.
In the present paper, as (\theta) we take the property of complete splitting of a group.
Definition ((^2)). A group (G) is called completely splittable if it can be represented as the set-theoretic sum of some set of its locally cyclic subgroups, intersecting pairwise in the identity subgroup.
Denoting by (r) the property of a group of being completely splittable, we arrive at the class of (\bar r J)-groups. Naturally, we consider only those (\bar r J)-groups which have (\bar r)-subgroups, and in what follows this will not be stated separately.
(\bar r)-Groups all of whose proper subgroups are (r)-groups were introduced by P. G. Kontorovich ((^3)) and called (z)-groups. All locally finite (z)-groups are finite; their description is given in ((^{3,4})). The question of the existence of infinite (z)-groups is open. It can be proved that an infinite (z)-group, if it exists, coincides with its commutant. Therefore (\bar r J)-groups have been studied under the additional assumption of local finiteness, and in the aperiodic case, of local solubility.
The class of (\bar r J)-groups obviously contains all (\bar c J)-groups, i.e. groups with invariant noncyclic subgroups. The structure of locally finite (\bar c J)-groups is known ((^{5-7})); therefore, in describing (\bar r J)-groups we do not present (\bar c J)-groups; for brevity, Dedekind groups are regarded as (\bar c J)-groups.
Notation: (|g|) is the order of the element (g); ([b,a]=bab^{-1}a^{-1}); (G') is the commutant of the group (G); (Z(G)) is the center of the group (G); (p,q) are distinct primes; (A\lambda B) is the semidirect product of the groups (A) and (B) with invariant factor (A) and complementary factor (B).
- All finite (z)-groups are soluble; consequently, all locally finite (\bar r J)-groups are soluble. Their structure is completely described by the three theorems below.
Theorem 1. A finite (p)-group is a (\bar r J)-group if and only if it is of one of the following types:
1) a $\bar c J$-group.
2) $G=G_1\times A$, $A$ is an elementary abelian $p$-group (possibly trivial), and
$G_1={a_0}\lambda{a_1}\lambda{a_2}\lambda\ldots\lambda{a_n}$, $n\geqslant 2$; $a_0^{p^u}=z$, $u\geqslant 1$, $|z|=|a_1|=\ldots=|a_n|=p$; $[a_i,a_{i-1}]=z$, $i=1,2,\ldots,n$; $[a_i,a_j]=1$ for $j\leqslant i-2$, $i=2,3,\ldots,n$.
3) $G=({a}\times{b})\lambda{c}$, $a^{p^{m-1}}=z$, $m\geqslant 1$, $|z|=|c|=p$, $|b|=p^2$, $[c,a]=1$, $[c,b]=z$ (in groups of types 1)—3) $p$ is an arbitrary prime number).
4) $G=({a}\lambda{b})\lambda{c}$, $a^p=z$, $|z|=|b|=|c|=p$, $p>2$, $[b,a]=z$, $[c,a]=b$, $[c,b]=1$.
5) $G=({a}\times{b})\lambda{c}$, $a^p=z$, $|z|=|b|=|c|=p$, $p>2$, $[c,a]=b$, $[c,b]=z^u$, where $u=1$ or $u=u_0$, the least quadratic nonresidue modulo $p$.
6) $G=({a}\times{b})\lambda{c}$, $a^p=z_1$, $b^p=z_2$, $|z_1|=|z_2|=|c|=p$, $p>2$, $[c,a]=z_2$, $[c,b]=z_1^u z_2^v$, where $4u+v^2$ is a quadratic nonresidue modulo $p$.
7) $G=(Q\times{b})\lambda{c}$, $|b|=4$, $|c|=2$, $Q$ is the quaternion group, $[c,g]=1$ for $g\in Q$, $[c,b]=z$, $z\in Q$, $|z|=2$.
8) $G=({a}\lambda{b})\lambda{c}$, $a^4=z$, $|z|=|b|=|c|=2$, $[b,a]=z$, $[c,a]=b$, $[c,b]=1$.
9) $H\subset G\cong G^$, $G^=H\lambda C$, $H={a}\times{b}$, $|a|=|b|=4$, $C={i}\times{j}$, $|i|=|j|=2$, and the elements $i,j$ induce on the group $H$ automorphisms defined by the following matrices over the field of four elements:
[
\begin{pmatrix}
-1&0\
0&-1
\end{pmatrix},
\quad
\begin{pmatrix}
-1&2\
2&1
\end{pmatrix}
\quad \text{or} \quad
\begin{pmatrix}
-1&0\
2&1
\end{pmatrix},
\quad
\begin{pmatrix}
-1&2\
0&1
\end{pmatrix}.
]
10) $G={a}\lambda{i}$, $|a|=8$, $|i|=2$, $[i,a]=a^2$.
11) $G=({a}\lambda{i})\lambda{j}$, $|a|=8$, $a^4=z$, $|i|=|j|=2$, $[i,a]=a^{-2}$, $[j,a]=z^{\varepsilon_1}$, $[j,i]=z^{\varepsilon_2}$, $\varepsilon_1,\varepsilon_2=0,1$, $\varepsilon_1+\varepsilon_2<2$.
Theorem 2. A finite nonprimary group $G$ is a $\bar r J$-group if and only if it is of one of the following types:
1) a $\bar c J$-group.
2) A Frobenius group of the form $C=({a}\times{b})\lambda{g}$, where $|a|=p$, $|b|=p^2$ or $|b|=pq$.
3) $G={a}\lambda({b}\lambda{c})$, $|a|=p>2$, $|b|=4$, $|c|=2$, $bab^{-1}=a^{-1}$, $ca=ac$, $cbc=b^{-1}$.
4) $G=({a}\times{d}\times{b})\lambda{c}$, $|a|=|d|=p>2$, $|b|=4$, $|c|=2$, $cac=a^{-1}$, $cdc=d^{-1}$, $cbc=b^{-1}$.
5) $G=({a}\lambda{b})\times({c}\lambda{d})$, $|a|=p$, $|b|=|d|=q$, $|c|=r$, $r$ is a prime, $r\ne q$, the subgroups ${a}\lambda{b}$ and ${c}\lambda{d}$ are noncyclic.
6) $G=[({a}\times{b})\lambda{d}]\times{c}$, where $({a}\times{b})\lambda{d}$ is a Frobenius group, $|a|=|b|=p$, $|d|=|c|=q$.
7) $G=({a}\lambda{b})\times B$, $|a|=p$, $|b|=q$, the subgroup ${a}\lambda{b}$ is noncyclic, $B$ is a group of exponent $q$, either abelian of rank $\geqslant 2$, or nonabelian of order $q^3$.
8) $G=({a}\times{b}\times{c})\lambda{d}$, $|a|=p$, $|b|=|c|=|d|=q$, the subgroup ${a}\lambda{d}$ is noncyclic, $db=bd$, $dcd^{-1}=bc$.
9) $G=P\lambda{g}$, $P$ is a Sylow $p$-subgroup of exponent $p$, and:
a) if $P$ is abelian, then $G={a}\times(P_1\lambda{g})$, the subgroup $P_1$ is noncyclic, and for any $g^k\ne 1$ there are no proper $g^k$-admissible subgroups in $P_1$;
b) if $P$ is nonabelian, then $P'=z(p)=Z(G)={a}$, and for any $g^k\ne 1$ there are no proper $g^k$-admissible subgroups in $P$ different from ${a}$.
We shall make a number of remarks about the proofs of Theorems 1 and 2.
The following is valid.
Proposition. The commutator subgroup of a finite $\bar r J$-group is contained in every one of its $z$-subgroups.
It follows from the description of finite (z)-groups, therefore, that the commutator subgroup of a finite primary (\bar rJ)-group is contained in a subgroup of order (p^3) with an element of order (p^2). It turns out that for (p\ne 2) the commutator subgroup may be either cyclic of order (p) (groups of types 2), 3) of Theorem 1) or abelian of type ((p,p)) (groups of types 4), 5), 6)). For (p=2) there also exist groups (types 9), 10)) with a cyclic commutator subgroup of order 4.
The proof of Theorem 2 is based on consideration of finite nonprimary (\bar rJ)-groups containing (z)-subgroups of all possible types, and on the description of finite solvable completely decomposable groups ((^9)).
It follows from Theorems 1 and 2 that an infinite locally finite (\bar rJ)-group has a simple commutator subgroup.
Theorem 3. An infinite locally finite group belongs to the class of (\bar rJ)-groups if and only if it is of one of the following types:
1) a (\bar cJ)-group.
2) (G=G_1\times A), where (A) is an elementary abelian (p)-group, and (G_1) is a central product over the subgroup ({a^p}) of some set of isomorphic groups of type ({a}\lambda{b}), (|a|=p^{\mu+1}), (\mu\ge 1), (|b|=p), ([b,a]=a^{1+p^\mu}), and, possibly, a cyclic group of order (p^{\mu+1}) or a group of type (p^\infty).
3) (G=(A\times{b})\lambda{c}), (A) is a group of type (p^\infty), (|b|=p^2), (|c|=p), ([c,a]=1) for (a\in A), ([c,b]=z), (|z|=p), (z\in A).
4) (G=({a}\lambda{b})\times B), (|a|=p), (|b|=q), (B) is an elementary abelian (q)-group of infinite rank, and ({a}\lambda{b}) is noncyclic.
Remark. In groups of type 2) of Theorem 1 and Theorem 3 every nonisolated subgroup is invariant (a subgroup (H) of a group (G) is isolated in (G) if (H) contains every cyclic subgroup of (G) with which it has a nontrivial intersection ((^8))). One can show that if a group (G) has a nonisolated subgroup and all nonisolated subgroups in (G) are invariant, then (G) is a group of the type indicated above.
- Theorem 4. A nonperiodic locally solvable (\bar rJ)-group contains a mixed abelian subgroup.
Theorem 4 follows from the following lemma:
Lemma. If in a torsion-free locally solvable group (G) all noninvariant subgroups are completely decomposable, then the group (G) itself is completely decomposable.
Theorem 5. Let (G) be a nonabelian group containing a mixed abelian subgroup. (G) is an (\bar rJ)-group if and only if its commutator subgroup is a cyclic subgroup of prime order (p), and all elements of finite order generate a locally cyclic (p)-subgroup or the quaternion group.
If, in addition, it is required that (G) be a group with a finite number of generators, then (G=G_1\times A), where (A) is a free abelian group of finite rank, and the subgroup (G_1) is no longer decomposable in a nontrivial way into a direct product. Defining relations have been found for the group (G_1).
Ural State University
named after A. M. Gorky
Sverdlovsk
Received
30 XII 1969
CITED LITERATURE
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(^2) P. G. Kontorovich, Mat. Sb., 5, 286 (1939).
(^3) P. G. Kontorovich, Mat. Sb., 7, 27 (1940).
(^4) A. I. Starostin, Uch. Zap. Ural. Univ., 23, 22 (1959).
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(^7) N. F. Liman, Mat. Zametki, 4, 1, 75 (1968).
(^8) P. G. Kontorovich, Mat. Sb., 19, 287 (1946).
(^9) A. N. Starostin, Izv. Vyssh. Uchebn. Zaved., Matematika, 2 (15), 168 (1960).