TY - JOUR

T1 - Equivalence of the Existence of Nonconjugate and Nonisomorphic Hall π-Subgroups

AU - Guo, W.

AU - Buturlakin, A. A.

AU - Revin, D. O.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - Let π be some set of primes. A subgroup H of a finite group G is called a Hall π-subgroup if any prime divisor of the order |H| of the subgroup H belongs to π and the index |G: H| is not a multiple of any number in π. The famous Hall theorem states that a solvable finite group always contains a Hall π-subgroup and any two Hall π-subgroups of such group are conjugate. The converse of the Hall theorem is also true: for any nonsolvable group G, there exists a set π such that G does not contain Hall π-subgroups. Nevertheless, Hall π-subgroups may exist in a nonsolvable group. There are examples of sets π such that, in any finite group containing a Hall π-subgroup, all Hall π-subgroups are conjugate (and, as a consequence, are isomorphic). In 1987 F. Gross showed that any set π of odd primes has this property. In addition, in nonsolvable groups for some sets π, Hall π-subgroups can be nonconjugate but isomorphic (say, in PSL 2 (7) for π = {2, 3}) and even nonisomorphic (in PSL 2 (11) for π = {2, 3}). We prove that the existence of a finite group with nonconjugate Hall π-subgroups for a set π implies the existence of a group with nonisomorphic Hall π-subgroups. The converse statement is obvious.

AB - Let π be some set of primes. A subgroup H of a finite group G is called a Hall π-subgroup if any prime divisor of the order |H| of the subgroup H belongs to π and the index |G: H| is not a multiple of any number in π. The famous Hall theorem states that a solvable finite group always contains a Hall π-subgroup and any two Hall π-subgroups of such group are conjugate. The converse of the Hall theorem is also true: for any nonsolvable group G, there exists a set π such that G does not contain Hall π-subgroups. Nevertheless, Hall π-subgroups may exist in a nonsolvable group. There are examples of sets π such that, in any finite group containing a Hall π-subgroup, all Hall π-subgroups are conjugate (and, as a consequence, are isomorphic). In 1987 F. Gross showed that any set π of odd primes has this property. In addition, in nonsolvable groups for some sets π, Hall π-subgroups can be nonconjugate but isomorphic (say, in PSL 2 (7) for π = {2, 3}) and even nonisomorphic (in PSL 2 (11) for π = {2, 3}). We prove that the existence of a finite group with nonconjugate Hall π-subgroups for a set π implies the existence of a group with nonisomorphic Hall π-subgroups. The converse statement is obvious.

KW - C condition

KW - conjugate subgroups

KW - Hall π-subgroup

KW - Hall -subgroup

KW - PRONORMALITY

KW - CONJUGACY

UR - http://www.scopus.com/inward/record.url?scp=85062513018&partnerID=8YFLogxK

U2 - 10.1134/S0081543818090109

DO - 10.1134/S0081543818090109

M3 - Article

AN - SCOPUS:85062513018

VL - 303

SP - 94

EP - 99

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

ER -