G. A. Malanina

Semidirect Products of Cyclic \(p\)-Groups

(Presented by Academician A. I. Mal'cev, 28 I 1960)

The work is devoted to \(p\)-groups that are decomposable into a semidirect product of cyclic subgroups. Groups of this kind can be characterized as \(p\)-groups possessing an invariant series with cyclic factors, all members of which are complemented in the group. In the commutative case this condition is satisfied by direct products of cyclic \(p\)-groups and only by them. In the general case the \(p\)-groups under consideration turn out to be two-step solvable (see Theorem 3).

It is known that in the commutative case of the \(p\)-groups considered here, any two of their direct decompositions with cyclic factors are isomorphic \((^{1})\); in the general case it turns out that this property is preserved for \(p \ne 2\) for any two semidirect decompositions of this kind (see Theorem 9).

It is known that every subgroup of a commutative group of the class of \(p\)-groups under consideration belongs to this class \((^{1})\). In the case of noncommutative \(p\)-groups decomposable into a semidirect product of cyclic subgroups, this property is lost (see Example 2); however, for abelian subgroups of such groups (for \(p \ne 2\)) it is nevertheless preserved (see Theorem 10).

In order to describe the class of \(p\)-groups decomposable into a semidirect product of cyclic subgroups, it proved necessary to define the external semidirect product of cyclic \(p\)-groups, i.e., the semidirect product of arbitrary cyclic \(p\)-groups under prescribed admissible relations between their elements (see Theorem 7). It turned out that the class of \(p\)-groups representable in the form of a semidirect product of cyclic subgroups coincides with the class of all possible external semidirect products of cyclic \(p\)-groups (see Theorem 8).

1. Definition 1. We shall call a group \(\mathfrak{G}\) the semidirect product of its subgroups \(\mathfrak{A}_1,\ldots,\mathfrak{A}_\alpha,\ldots\) \((\alpha \in \mathfrak{M}, \mathfrak{M}\)—some set of ordinal numbers \(\nu\) satisfying one of two conditions: \(1 \le \nu < \gamma\) or \(1 \le \nu \le \gamma)\) and shall write

\[ \mathfrak{G}=[\mathfrak{A}_1\ldots \mathfrak{A}_\alpha\ldots]\quad(\alpha \in \mathfrak{M}), \tag{1} \]

if the following requirements are fulfilled: 1) the group \(\mathfrak{G}\) is generated by the subgroups \(\mathfrak{A}_1,\ldots,\mathfrak{A}_\alpha,\ldots\); 2) if \(\alpha\) is an arbitrary ordinal number from \(\mathfrak{M}\) distinct from one, then the subgroup \(\mathfrak{G}_\alpha\), generated by the subgroups \(\mathfrak{A}_\beta\), \(\beta < \alpha\) \((\beta \in \mathfrak{M})\), is invariant in the group \(\mathfrak{G}\) and has trivial intersection with the subgroup \(\mathfrak{G}^{(\alpha)}\), generated by the subgroups \(\mathfrak{A}_\nu\), \(\nu \ge \alpha\) \((\nu \in \mathfrak{M})\).

We shall call the group \(\mathfrak{G}\) decomposable into a semidirect product of its subgroups \(\mathfrak{A}_\alpha\) \((\alpha \in \mathfrak{M})\), and these subgroups \(\mathfrak{A}_\alpha\) the factors of the given decomposition.

In the paper \((^{2})\) some properties of groups decomposable into a semidirect product of their subgroups, following from the above definition, are noted, and, in particular, the following: a) all factors of the decomposition (1) are pairwise permutable with one another; b) the intersection of any subgroup

\(\mathfrak A_\alpha\) \((\alpha\in\mathfrak M)\) with the subgroup generated by all \(\mathfrak A_\beta,\ \beta\ne\alpha\) \((\beta\in\mathfrak M)\), is equal to the identity;

c) for any two factors \(\mathfrak A_\alpha\) and \(\mathfrak A_\beta,\ \alpha<\beta\), of the decomposition (1), \(\{\mathfrak A_\alpha,\mathfrak A_\beta\}=[\mathfrak A_\alpha\mathfrak A_\beta]\).

Theorem 1. If \(\mathfrak G=[\{A\}\{B\}]\) is some decomposition of a \(p\)-group \(\mathfrak G\) into a semidirect product of cyclic subgroups with relations

\[ A^{p^m}=B^{p^n}=E,\qquad BAB^{-1}=A^s,\qquad A^s\ne A, \]

then its commutator subgroup coincides with the subgroup \(\{A^{s-1}\}\); this subgroup is distinct from \(\{A\}\), and therefore \(s=Np^k+1\), where \((N,p)=1,\ 1\le k<m\).

Lemma 1. If

\[ \mathfrak G=[\{A_1\}\ldots\{A_\alpha\}\ldots]\qquad(\alpha\in\mathfrak M) \tag{2} \]

is a decomposition of the \(p\)-group \(\mathfrak G\) into a semidirect product of cyclic subgroups with relations

\[ A_\alpha^{p^{m_\alpha}}=E,\qquad A_\beta A_\alpha A_\beta^{-1}=A_\alpha^{s_{\alpha\beta}},\qquad \alpha<\beta\ (\beta\in\mathfrak M), \tag{3} \]

where \(s_{\alpha\beta}=N_{\alpha\beta}p^{k_{\alpha\beta}}+1\) \(\bigl((N_{\alpha\beta},p)=1,\ k_{\alpha\beta}\ge1\ \text{or}\ N_{\alpha\beta}=0\bigr)\), then the congruences

\[ s_{\alpha\beta}^{p^{m_\beta}}\equiv1\pmod {p^{m_\alpha}},\qquad s_{\alpha\beta}^{\,s_{\beta\gamma}-1}\equiv1\pmod {p^{m_\alpha}},\qquad \beta<\gamma\ (\gamma\in\mathfrak M) \]

are satisfied.

If the subgroup \([\{A_\alpha\}\{A_\beta\}]\) of the decomposition (2) is abelian, then we shall set the number \(k_{\alpha\beta}\) in the relations (3) equal to \(m_\alpha\).

Theorem 2. The lower layer \(\mathfrak G_1\) of the \(p\)-group \(\mathfrak G\) (i.e. the set of its elements of order not exceeding \(p\)) with decomposition (2) and relations (3), for \(p\ne2\), coincides with the subgroup \(\{A_1^{p^{m_1-1}}\}\times\cdots\times\{A_\alpha^{p^{m_\alpha-1}}\}\times\cdots\).

Remark. For \(2\)-groups of this kind, Theorem 2 does not hold.

Example 1. In the group \(\mathfrak G=[\{A\}\{B\}]\) with relations \(A^8=B^2=E,\ BAB^{-1}=A^3\) (the existence of such a group follows from Theorem 7 given below), the lower layer consists of its following 6 elements: \(E,\ A^4,\ B,\ A^2B,\ A^4B,\ A^6B\). If Theorem 2 were valid in the case of the group under consideration, then its lower layer would have to contain only the elements \(E,\ A^4,\ B,\ A^4B\).

2. Theorem 3. The commutator subgroup \(\mathfrak G'\) of the \(p\)-group \(\mathfrak G\) with decomposition (2) and relations (3) coincides with the subgroup
\[ \{A_1^{p^{\bar k_1}}\}\times\cdots\times\{A_\alpha^{p^{\bar k_\alpha}}\}\times\cdots, \]
where \(\bar k_\alpha=\min(k_{\alpha\alpha+1},\ldots,k_{\alpha\beta},\ldots),\ \alpha<\beta\), and each subgroup \(\{A_\alpha^{p^{\bar k_\alpha}}\}\) is invariant in the group \(\mathfrak G\).

Theorem 4. If

\[ \mathfrak G=\mathfrak G_0\supseteq\mathfrak G_1\supseteq\mathfrak G_2\supseteq\cdots\supseteq\mathfrak G_n\supseteq\cdots \]

is the lower central series of the \(p\)-group \(\mathfrak G\) with decomposition (2) and relations (3), then
\[ \mathfrak G_n=\{A_1^{p^{\bar k_1 n}}\}\times\cdots\times\{A_\alpha^{p^{\bar k_\alpha n}}\}\times\cdots,\qquad n=0,1,2,\ldots, \]
where \(\bar k_\alpha=\min(k_{\alpha\alpha+1},\ldots,k_{\alpha\beta},\ldots),\ \beta>\alpha\ (\beta\in\mathfrak M)\).

Theorem 5. A \(p\)-group \(\mathfrak G\) with decomposition (2) and relations (3) is nilpotent if and only if for the numbers \(k_1,k_2,\ldots,\bar k_\alpha,\ldots\), where \(k_\alpha=\min(k_{\alpha\alpha+1},\ldots,k_{\alpha\beta},\ldots),\ \beta>\alpha\ (\beta\in\mathfrak M)\), there exists a natural number \(M\) such that for every \(\alpha\) the inequality \(m_\alpha\le\bar k_\alpha M\) holds.

Corollary 1. If the orders of all factors of the decomposition (2) of the \(p\)-group \(\mathfrak G\) are bounded in the aggregate, then the group \(\mathfrak G\) is nilpotent.

3. Theorem 6. A mapping \(\varphi\) of the \(p\)-group \(\mathfrak G\) with decomposition (2) and relations (3) onto itself, defined by the following two conditions, is an automorphism of this group:

1) for all elements \(A_\alpha^x\) of \(\{A_\alpha\}\) \((\alpha \in \mathfrak M)\)

\[ A_\alpha^x\varphi=A_\alpha^{x r_{\alpha\varphi}}, \]

where \(r_{\alpha\varphi}\) is some natural number of the form \(r_{\alpha\varphi}=M_{\alpha\varphi}p^{q_{\alpha\varphi}}+1\), with \((M_{\alpha\varphi},p)=1,\ q_{\alpha\varphi}\ge 1\), or \(M_{\alpha\varphi}=0\), satisfying the relation

\[ s_{\gamma\alpha}^{\,r_{\alpha\varphi}-1}\equiv 1\pmod {p^{m_\gamma}},\qquad \gamma<\alpha\ (\gamma\in \mathfrak M); \]

2) for every element \(g=A_{\beta_1}^{i_1}A_{\beta_2}^{i_2}\cdots A_{\beta_t}^{i_t}\in\mathfrak G,\quad \beta_1<\beta_2<\cdots<\beta_t\) \((\beta_1,\ldots,\beta_t\subset\mathfrak M)\)

\[ g\varphi=(A_{\beta_1}^{i_1})\varphi\,(A_{\beta_2}^{i_2})\varphi\cdots(A_{\beta_t}^{i_t})\varphi . \]

Theorem 7. Let \(\{A_\alpha\}\) \((\alpha\in\mathfrak M)\) be arbitrary cyclic \(p\)-groups of orders \(p^{m_\alpha}\), and let the numbers \(s_{\alpha\beta}\), \(\beta>\alpha\) \((\beta\in\mathfrak M)\), satisfy the conditions:

1) \(s_{\alpha\beta}=N_{\alpha\beta}p^{k_{\alpha\beta}}+1\), where \((N_{\alpha\beta},p)=1,\ k_{\alpha\beta}\ge 1\), or \(N_{\alpha\beta}=0\);

2) \(s_{\alpha\beta}^{\,s_{\beta\gamma}-1}\equiv 1\pmod {p^{m_\alpha}},\qquad \gamma>\beta\ (\gamma\in\mathfrak M)\);

3) \(s_{\alpha\beta}^{\,p^{m_\beta}}\equiv 1\pmod {p^{m_\alpha}}\).

Then the group \(\mathfrak G\) (we shall call it an outer semidirect product of the cyclic \(p\)-groups \(\{A_\alpha\}\)), generated by the elements of all groups \(\{A_\alpha\}\) subject to the relations

\[ A_\beta A_\alpha A_\beta^{-1}=A_\alpha^{s_{\alpha\beta}},\qquad \alpha<\beta \tag{4} \]

(the relations between the elements of each group \(\{A_\alpha\}\) are preserved also in the group \(\mathfrak G\)), decomposes into the semidirect product

\[ \mathfrak G=[\{A_1\}\cdots\{A_\alpha\}\cdots], \]

satisfying the relations (4).

Remark. In view of a certain arbitrariness in the choice of the numbers \(s_{\alpha\beta}\), for one and the same cyclic \(p\)-groups \(\{A_\alpha\}\) \((\alpha\in\mathfrak M)\) one can construct a series of outer semidirect products that are not isomorphic to one another.

From Theorem 7 and Lemma 1 one can derive the following proposition.

Theorem 8. The class of \(p\)-groups decomposable into a semidirect product of cyclic subgroups coincides with the class of all possible outer semidirect products of cyclic \(p\)-groups.

4. Definition 2. Two decompositions of a \(p\)-group into a semidirect product of cyclic subgroups will be called isomorphic if between their factors one can establish such a one-to-one correspondence under which the corresponding factors are isomorphic groups.

Theorem 9. If a \(p\)-group \(\mathfrak G\) is decomposable into a semidirect product of cyclic subgroups, then for \(p\ne 2\) any two such decompositions of it are isomorphic to one another.

Theorem 10. Every abelian subgroup of a \(p\)-group \(\mathfrak G\), decomposable into a semidirect product of cyclic subgroups, for \(p\ne 2\), is decomposable into a direct product of cyclic subgroups.

Remark 1. We shall call a noncommutative \(2\)-group \(\mathfrak G\) with a decomposition (2) a special group if in this decomposition there are such factors \(\{A_\alpha\}\) and \(\{A_\beta\}\), \(\beta>\alpha\) \((\beta\in\mathfrak M)\), that the elements \(A_\alpha\) and \(A_\beta\) of the noncommutative group \([\{A_\alpha\}\{A_\beta\}]\) are connected by the relation \(A_\beta A_\alpha A_\beta^{-1}=A_\alpha^{N\cdot 2+1}\), where \((N,2)=1\). Theorems 2, 9, and 10 are also valid for \(2\)-groups with decomposition (2) when they are not special. The case of special groups requires additional investigations.

Remark 2. Not every subgroup of a \(p\)-group \((p\ne 2)\) with decomposition (2) is decomposable into a semidirect product of cyclic subgroups.

Example 2. In the \(p\)-group \(\mathfrak{G}=[\{A\}\{B\}]\) with relations \(A^{p^5}=B^{p^4}=E\) \((p\ne 2)\) and \(BAB^{-1}=A^{p^2+1}\) (the existence of such a group follows from Theorem 7), the subgroup \(\mathfrak{A}=\{A^{p^2}\}\{AB\}\) does not decompose into a semidirect product of cyclic subgroups.

The topic of the present work was proposed to the author by S. N. Chernikov and was developed under his supervision.

Perm State University
named after A. M. Gorky

Received
26 I 1960

References Cited

\({}^{1}\) A. G. Kurosh. Group Theory, 2nd ed., Moscow—Leningrad, 1953. \({}^{2}\) G. S. Shevtsov, Izv. vyssh. uchebn. zaved., Matem., 1, 184 (1958).