MATHEMATICS

R. P. IGNAT’EVA

THEOREMS ON THE EXISTENCE AND EMBEDDING OF SUBGROUPS IN A FINITE GROUP

(Presented by Academician A. I. Mal’cev on 23 IV 1960)

§ 1. In the present paper we consider questions of the existence and embedding of subgroups in a finite group in connection with the notions introduced by S. A. Chunikhin in \((^{1,2})\): those of a block, separable, and reduced \(\Pi\)-block divisor of the order of a group, and with Theorem 14 obtained by him in \((^1)\).

The terminology and notation are taken from works \((^{1-3})\).

§ 2. Lemma 1. Let \(s\) be a separable divisor of the order \(g\) of a group \(G\), and let \(A\) be a subgroup of order \(a\) of the group \(G\). Then \((s,a)\) is a separable divisor of the order \(a\) of the subgroup \(A\).

Lemma 2. If \(N\) is an invariant subgroup of order \(n\) of a group \(G\), and the number \(s\) is a separable divisor of the order \(g\) of the group \(G\), then the number

\[ \frac{s}{(s,n)}=d \]

is a separable divisor of the order \(\frac{g}{n}\) of the factor group \(G/N\).

Theorem 1. Let \(h'\) be a reduced \(\Pi\)-block divisor, and \(s\) a separable divisor of the order \(g\) of a group \(G\). Then the group \(G\) has at least one subgroup of order

\[ \frac{h's}{(h',s)}\,c, \]

where \(c\) is a \(\Pi\)-prime number.

§ 3. Definition. A divisor \(hs\) of the order \(g\) of a group \(G\) will be called a block-separable divisor if it is the product of a block divisor \(h\) and a separable divisor \(s\) of the order \(g\) of the group \(G\).

Lemma 3. If \(hs\) is a block-separable divisor of the order \(g\) of a group \(G\), and if \(a\) is the order of a subgroup \(A\) of the group \(G\), then \((hs,a)\) is a block-separable divisor of the order \(a\) of the subgroup \(A\).

Lemma 4. Let \(N\) be an invariant subgroup of order \(n\) of a group \(G\), and let the number \(hs\) be a block-separable divisor of the order \(g\) of the group \(G\). Then the number

\[ \frac{hs}{(hs,n)} \]

is a block-separable divisor of the order \(\frac{g}{n}\) of the factor group \(G/N\).

Theorem 2. If a divisor \(hs\) of the order \(g\) of a group \(G\) is block-separable, then every soluble subgroup \(A\) of order \(a\) dividing \(hs\) is contained in at least one of the subgroups of order \(hs\) of the group \(G\).

§ 4. Lemma 5. If \(h\) is a block divisor and \(s\) is a separable divisor of the order \(g\) of a group \(G\), then their greatest common divisor \((h,s)\) is a separable divisor of the order \(g\) of the group \(G\).

Theorem 3. Let

\[ h_1,h_2,\ldots,h_n \tag{1} \]

be some nonempty set of block divisors, and

\[ s_1,s_2,\ldots,s_m \tag{2} \]

some nonempty set of separable divisors of the order \(g\) of the group \(G\). Then for any two nonempty subsets \(h_{i_1}, h_{i_2}, \ldots, h_{i_r}\) and

\(h_{i_1}, h_{i_2}, \ldots, h_{i_l}\) of the set (1), in the group \(G\) there exist at least one subgroup of each of the orders:

\[ A=(h_{j_1},s_{k_1})\left(\frac{h_{j_2}}{(h_{j_2},h_{j_1})},s_{k_2}\right)\cdots \left(\frac{h_{j_l}}{(h_{j_l},h_{j_1}h_{j_2}\cdots h_{j_{l-1}})},s_{k_l}\right), \]

\[ B=\frac{(g,h_{i_1}h_{i_2}\cdots h_{i_r})\cdot A} {((g,h_{i_1}h_{i_2}\cdots h_{i_r}),A)}, \]

where each of the numbers \(s_{k_1}, s_{k_2}, \ldots, s_{k_l}\) is equal to one of the proper divisors of the set (2), and all subgroups of each of the orders \(A\) are conjugate to one another in the group \(G\), while those subgroups of order \(B\) for which

\[ \frac{A}{((g,h_{i_1}h_{i_2}\cdots h_{i_r}),A)}=d>1 \]

are \(\Pi(d)\)-solvable.

Theorem 4. If \(h_1\) and \(h_2\) are relatively prime divisors, and \(s\) is a proper divisor of the order \(g\) of the group \(G\), then \(G\) has at least one subgroup of each of the orders:

\[ a_1=\frac{h_1}{d}(d,s),\qquad a_2=\frac{h_2}{d}(d,s),\qquad a_3=\frac{d(h_1,s)}{(d,s)}, \]

\[ a_4=\frac{d(h_2,s)}{(d,s)}, \]

\[ a_5=\frac{h_1(h_2,s)}{(d,s)},\qquad a_6=\frac{h_2(h_1,s)}{(d,s)},\qquad a_7=\frac{h_1}{d}(h_2,s), \]

\[ a_8=\frac{h_2}{d}(h_1,s),\qquad a_9=\frac{h_1h_2}{d^2}(d,s),\qquad a_{10}=d\left(\frac{h_1h_2}{d^2},s\right), \]

where \(d=(h_1,h_2)\), and:

if \((d,s)=m_1>1\), subgroups of orders \(a_1,a_2,a_9\) are \(\Pi(m_1)\)-solvable;

if \(\dfrac{(h_1,s)}{(d,s)}=m_2>1\), subgroups of orders \(a_3,a_6\) are \(\Pi(m_2)\)-solvable;

if \(\dfrac{(h_2,s)}{(d,s)}=m_3>1\), subgroups of orders \(a_4,a_5\) are \(\Pi(m_3)\)-solvable;

if \((h_2,s)=m_4>1\), a subgroup of order \(a_7\) is \(\Pi(m_4)\)-solvable;

if \((h_1,s)=m_5>1\), a subgroup of order \(a_8\) is \(\Pi(m_5)\)-solvable;

if \(\left(\dfrac{h_1h_2}{d^2},s\right)=m_6>1\), a subgroup of order \(a_{10}\) is \(\Pi(m_6)\)-solvable.

Theorem 5. If

\[ h'_1,h'_2,\ldots,h'_n \tag{3} \]

is the set of reduced \(\Pi\)-relatively prime divisors, and

\[ s_1,s_2,\ldots,s_m \tag{4} \]

is the set of proper divisors of the order \(g\) of the group \(G\), then for any two nonempty subsets \(h'_{i_1}, h'_{i_2}, \ldots, h'_{i_r}\) and \(h'_{j_1}, h'_{j_2}, \ldots, h'_{j_l}\) of the set (3), in the group \(G\) there exists at least one subgroup of each of the orders:

\[ A=(h'_{j_1},s_{k_1}) \left(\frac{h'_{j_2}}{(h'_{j_2},h'_{j_1})},s_{k_2}\right)\cdots \left(\frac{h'_{j_l}}{(h'_{j_l},h'_{j_1}h'_{j_2}\cdots h'_{j_{l-1}})},s_{k_l}\right), \]

\[ B= \frac{(g,h'_{i_1}h'_{i_2}\cdots h'_{i_r})\cdot A} {((g,h'_{i_1}h'_{i_2}\cdots h'_{i_r}),A)}\,c, \]

where each of the numbers \(s_{k_1}, s_{k_2}, \ldots, s_{k_l}\) is equal to one of the numbers of the set (4), and \(c\) is a \(\Pi\)-prime number; moreover, all subgroups of each of the orders \(A\) are conjugate to one another in the group \(G\).

§ 5. In conclusion, we note that from Theorem 2, for \(s = 1\), there follows Theorem 5 of paper \((^1)\), and for \(h = 1\)—the theorem on the embedding of subgroups of a \(\Pi\)-separable group, D5.2 of paper \((^5)\).

If in Theorem 1 the set \(\Pi\) contains all distinct prime divisors of the order of the group, then from it one obtains Theorem 14 of paper \((^1)\). As is known, from the 14th theorem of S. A. Chunikhin there follow, as its special cases, the theorems on the existence of subgroups: those of Sylow, P. Hall (for solvable groups), Schur (on factorization of groups), S. A. Chunikhin (Theorem 1 of \((^1)\)) and the fundamental theorem on \(\Pi\)-separable groups \((^3)\).

Theorem 3 is a special case of Theorem 5, when the set \(\Pi\) contains all distinct prime divisors of the order of the group.

The author expresses his deep gratitude to S. A. Chunikhin for suggesting the topic and for his unfailing interest in the work.

Kabardino-Balkarian
State University

Received
20 IV 1960

REFERENCES

\(^1\) S. A. Chunikhin, Matem. sborn., 39 (81), No. 3, 465 (1956).
\(^2\) S. A. Chunikhin, Matem. sborn., 43 (85), No. 1, 49 (1957).
\(^3\) S. A. Chunikhin, DAN, 59, No. 3, 443 (1948).
\(^4\) S. A. Chunikhin, DAN, 119, No. 5, 888 (1958).
\(^5\) P. Hall, Proc. Lond. Math. Soc., (3), 6, No. 22, 286 (1956).
\(^6\) O. Yu. Schmidt, Abstract Theory of Groups, 1933.