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UDC 519.45
MATHEMATICS
D. A. SUPRUNENKO
ON LOCALLY NILPOTENT SUBGROUPS OF THE INFINITE SYMMETRIC GROUP
(Presented by Academician A. I. Mal’cev on 19 VI 1965)
Let \(X\) be an arbitrary nonempty set; \(S(X)\) the group of all bijective mappings \(f: X \to X\); \(SF(X)\) the subgroup of \(S(X)\) consisting of all such \(f \in S(X)\) that \(f(x) \ne x\) only for finitely many points \(x \in X\).
In the present paper it is proved that, in the case of infinite \(X\), every transitive locally nilpotent subgroup of the group \(SF(X)\) is a \(p\)-group. Nilpotent subgroups of the group \(SF(X)\) are described. One property of stationary subgroups of transitive nilpotent permutation groups of finite degree is given.
I. If \(\mathfrak G\) is an arbitrary group, then every homomorphism \(u: \mathfrak G \to S(X)\) will be called a representation of the group \(\mathfrak G\). Two representations \(u: \mathfrak G \to S(X)\) and \(v: \mathfrak G \to S(Y)\) are called equivalent if there exists a bijective mapping \(\varphi: X \to Y\) such that, for every \(a\) in \(\mathfrak G\),
\(v(a)=\varphi u(a)\varphi^{-1}\). Let \(\mathfrak H\) be an intransitive subgroup of \(S(X)\), and let \(X_\alpha\) be one of the systems of imprimitivity of the group \(\mathfrak H\). We introduce the transitive representation \(r_\alpha\) of the group \(\mathfrak H\) by putting
\[ r_\alpha:\mathfrak H \to S(X_\alpha), \qquad r_\alpha(h)=h|X_\alpha, \tag{1} \]
where \(h|X_\alpha\) is the restriction of \(h\) to \(X_\alpha\), \(h \in \mathfrak H\).
Let \(\mathfrak G\) be a transitive subgroup of the group \(S(X)\); let \(\mathfrak H\) be a subgroup of its centralizer \(C(\mathfrak G)\) in \(S(X)\). Then the following two lemmas are valid.
Lemma 1. If the group \(\mathfrak H\) is transitive, then on the set \(X\) one can define a group operation \(+\) such that the group \(\mathfrak G\) turns out to be the left regular representation of the group \(\langle X,+\rangle\), and \(\mathfrak H\) the right regular representation of the group \(\langle X,+\rangle\).
Lemma 2. If \(\mathfrak H\) is intransitive, and \(X=\bigcup_{\alpha\in I} X_\alpha\) is the decomposition of \(X\) into systems of imprimitivity of \(\mathfrak H\), then for any \(\alpha,\beta \in I\) the transitive representations \(r_\alpha, r_\beta\) of the group \(\mathfrak H\), defined by formulas (1), are equivalent.
From Lemma 2 it follows:
Lemma 3. If \(X\) is an infinite set, and \(\Gamma\) is a transitive subgroup of the group \(SF(X)\), then the center of the group \(\Gamma\) coincides with the identity group.
II. Lemma 4. Let \(X\) be an arbitrary nonempty set, and let \(\mathfrak P\) be a transitive \(p\)-subgroup of the group \(SF(X)\). If a locally nilpotent subgroup \(\Gamma\) of the group \(SF(X)\) contains \(\mathfrak P\), then \(\Gamma\) is also a \(p\)-group. In particular, a transitive Sylow \(p\)-subgroup of the group \(SF(X)\) is a maximal locally nilpotent subgroup of the group \(SF(X)\).
Proof. For finite \(X\) the lemma was proved in the paper \((^1)\). Let \(X\) be an infinite set, and let \(\Gamma\) be a locally nilpotent subgroup of \(SF(X)\) containing \(\mathfrak P\). Suppose further, contrary to the assertion of the lemma, that \(\Gamma\) is not a \(p\)-group. Then, by virtue of the periodicity of \(\Gamma\), in \(\Gamma\) there is an element \(a\) of prime order \(q \ne p\). Since \(\Gamma\) is locally nilpotent, the cyclic group \(A=(a)\) is contained in the centralizer of the group \(\mathfrak P\) in
\(S(X)\). If now \(X=\bigcup_{\alpha\in I} X_\alpha\) is the decomposition of \(X\) into the systems of imprimitivity of the group \(A\), then, according to Lemma 2, the representations \(r_\alpha:A\to S(X_\alpha)\), where \(r_\alpha(a)=a/X_\alpha\), are pairwise equivalent. Only two cases are possible: either each \(X_\alpha\) consists of \(q\) points, permuted cyclically by \(a/X_\alpha\), or each \(X_\alpha\) consists of one point. In the first case \(a\) leaves fixed no point of the set \(X\), which contradicts the inclusion \(a\in SF(X)\); in the second, \(a\) leaves every point of the set \(X\) fixed. The latter contradicts the fact that the order of \(a\) is equal to the prime number \(q\). The lemma follows.
Theorem 1. Let \(X\) be an infinite set*. Then every transitive locally nilpotent subgroup of the group \(SF(X)\) is a \(p\)-group. In particular, a maximal transitive locally nilpotent subgroup of \(SF(X)\) is a Sylow \(p\)-subgroup of the group \(SF(X)\).
Proof. Let \(\Gamma\) be a transitive locally nilpotent subgroup of \(SF(X)\). If one of the Sylow \(p\)-subgroups of the group \(\Gamma\) is transitive, then the theorem follows from Lemma 4. We first show that the number of systems of imprimitivity of a Sylow \(p\)-subgroup of the group \(\Gamma\) is finite. Let \(\mathcal H\) be a nonidentity Sylow \(p\)-subgroup of the group \(\Gamma\), and let \(X=\bigcup_{\alpha\in I}X_\alpha\) be the decomposition of \(X\) into the systems of imprimitivity of the group \(\mathcal H\). Consider the transitive representations \(r_\alpha\) of the group \(\mathcal H\)
\[ r_\alpha:\mathcal H\to S(X_\alpha),\qquad r_\alpha(h)=h_\alpha=h/X_\alpha,\quad h\in\mathcal H. \tag{2} \]
For any \(\alpha\) and \(\beta\) in \(I\), the representations \(r_\alpha\) and \(r_\beta\) are equivalent. Indeed, \(\Gamma\) can be represented as the direct product
\[ \Gamma=\mathcal H\mathcal D, \tag{3} \]
where \(\mathcal D\) is a normal divisor of \(\Gamma\), the orders of whose elements are relatively prime to \(p\). Since \(\Gamma\) is transitive, by virtue of (3), for any \(\alpha,\beta\in I\) we have \(X_\beta=d(X_\alpha)\), where \(d\in\mathcal D\). Then for the restriction \(h_\beta=h/X_\beta\), where \(h\in\mathcal H\), one can write
\(h_\beta(x_\beta)=h(x_\beta)=hd(x_\alpha)=dh(x_\alpha)=dhd^{-1}(x_\beta)=d_\alpha h_\alpha d_\alpha^{-1}(x_\beta)\), where \(x_\alpha\in X_\alpha\), \(x_\beta\in X_\beta\), \(d_\alpha=d/X_\alpha\). Hence \(h_\beta=d_\alpha h_\alpha d_\alpha^{-1}\), \(r_\beta(h)=d_\alpha r_\alpha(h)d_\alpha^{-1}\). The equivalence of \(r_\alpha\) and \(r_\beta\) is proved. From the pairwise equivalence of the representations \(r_\alpha\) and the inclusion \(\mathcal H\subset SF(X)\), it follows that the set of systems of imprimitivity of the group \(\mathcal H\) is finite. Hence it follows in turn that every nonidentity Sylow \(p\)-subgroup of the group \(\Gamma\) is infinite.
We now show that a nonidentity Sylow \(p\)-subgroup \(\mathcal P\) of the group \(\Gamma\) is transitive. Suppose \(\mathcal P\) is intransitive. Then \(X\) is decomposed into a finite number of systems of imprimitivity of the group \(\mathcal P\):
\[ X_1\cup\ldots\cup X_k=X,\qquad k>1. \tag{4} \]
Since \(\mathcal P\) is a normal divisor of \(\Gamma\), (4) is a decomposition of \(X\) into systems of imprimitivity of \(\Gamma\).
Consider now the homomorphism \(\gamma:\Gamma\to S_k\), where \(S_k\) is the symmetric group of degree \(k\), permuting the systems \(X_1,\ldots,X_k\) among themselves.
Let \(N\) be the kernel of the homomorphism \(\gamma\). We show that \(N=\mathcal P\). Obviously, \(\Gamma\) is represented as the direct product \(\Gamma=\mathcal P U\), where the orders of the elements of the direct factor \(U\) are relatively prime to \(p\). It is clear that \(N\) contains \(\mathcal P\). Consequently, it is enough to show that \(N\) has no nonidentity element \(u\) from \(U\). Let \(u\in N\cap U\). Then for \(j=1,\ldots,k\), \((u_j)\mathcal P_j\), where \(u_j=u/X_j\), \(\mathcal P_j=\mathcal P/X_j\), is a locally nilpotent transitive subgroup of the group \(SF(X_j)\), containing the transitive \(p\)-group \(\mathcal P_j\). According to Lemma 4, \((u_j)\mathcal P_j\) is also a \(p\)-group. Consequently, \(u\) is a \(p\)-element. Since \(U\) has no nonidentity \(p\)-elements, \(u=1\), \(N=\mathcal P\). From the equality \(N=\mathcal P\) it follows that \(U\) is a finite subgroup of \(\Gamma\). Consequently, every Sylow \(q\)-subgroup of the group \(\Gamma\) contained in \(U\) is finite. But above
* As follows from paper (1), in the case of finite \(X\) Theorem 1 is false.
we have proved that in \(\Gamma\) there are no nonidentity finite Sylow \(q\)-subgroups. Thus the assumption that \(\mathscr G\) is intransitive leads to a contradiction. Hence \(\mathscr G\) is transitive. The theorem follows from this.
Corollary. Let \(\mathscr G\) be a maximal locally nilpotent subgroup of the group \(SF(X)\), where \(X\) is an infinite set. Then \(\mathscr G\) is the direct product of its restrictions \(\mathscr G_\alpha\) to the transitivity systems \(X_\alpha\) of the group \(\mathscr G\). For infinite \(X_\alpha\) the group \(\mathscr G_\alpha\) is a Sylow \(p\)-subgroup of the group \(SF(X_\alpha)\), and for finite \(X_\alpha\), \(\mathscr G_\alpha\) is a maximal transitive nilpotent subgroup of \(S(X_\alpha)\), described in (1).
The Sylow \(p\)-subgroups of \(SF(X)\) for countable \(X\) were studied by I. D. Ivanyuta in the paper (3).
III. From Lemma 3 and the results of paper (2) it follows
Theorem 2. Let \(X\) be an infinite set, and let \(\mathscr G\) be a nilpotent subgroup of the group \(SF(X)\). Then \(\mathscr G\) is intransitive, and every one of its transitivity systems \(X_\alpha\) is finite. \(\mathscr G\) is maximal among the nilpotent subgroups of \(SF(X)\) if and only if the following four conditions hold simultaneously:
1) \(\mathscr G\) is the direct product of its restrictions \(\mathscr G_\alpha = \mathscr G / X_\alpha\) to the transitivity systems \(X_\alpha\);
2) \(\mathscr G_\alpha\) is a maximal transitive nilpotent subgroup of the group \(S(X_\alpha)\);
3) the number of transitivity systems of one and the same order \(p^t\), where \(p\) is a prime number and \(t > 0\), is less than the number \(p\);
4) among the transitivity systems of the group \(\mathscr G\) there is at most one system consisting of a single point.
Analogously, with the aid of Lemma 3, one can obtain
Proposition. Every \(ZA\)-group contained in \(SF(X)\) is nilpotent.
IV. We give one property of nilpotent transitive subgroups of the finite symmetric group \(S_n\) of degree \(n\).
If \(n\) is a natural number, then denote by the letter \(m\) the product of all distinct prime divisors of the number \(n\).
Theorem 3. Let \(\Gamma\) be a maximal transitive nilpotent subgroup of \(S_n = S(X)\). If \(\Gamma_1\) is the stabilizer subgroup of the group \(\Gamma\); \(Y\) is the set of all such \(x\) in \(X\) that \(g_1(x)=x\) for every substitution \(g_1 \in \Gamma_1\), then the number of points of the set \(Y\) coincides with the number \(m\).
Corollary. Let \(\Gamma\) be a transitive nilpotent subgroup of the group \(S_n = S(X)\). Then a substitution \(g\) from \(\Gamma\) that leaves one point \(x \in X\) fixed leaves fixed at least \(m\) points.
Institute of Mathematics
Academy of Sciences of the BSSR
Received
10 VI 1965
REFERENCES
- D. A. Suprunenko, DAN, 99, 23 (1954).
- D. A. Suprunenko, G. P. Zhavrid, Dokl. BSSR, 11, No. 8, 320 (1958).
- I. D. Ivanyuta, Ukr. Mat. Zh., 15, No. 3, 240 (1963).