A341048 Numbers m such that there is a group of order m that is not supersolvable (NSS) but "converse Lagrange theorem" (CLT).

224, 2464, 2912, 3159, 3808, 4256, 5152, 6318, 6496, 8288, 9184, 9632

Offset

1,1

Comments

The converse to Lagrange's theorem does not hold. A340511 lists the numbers n such that there exists a group of order n which has no subgroup of order d, for some divisor d of n; they are called "non-converse Lagrange theorem" (NCLT) orders.

A finite group is supersolvable (SS) if it has a normal series of subgroups with cyclic factors; A066085 lists the numbers for which there exists a group of order n that is not supersolvable; they are called a "non-supersolvable" (NSS) order.

Theorem: Every NCLT order is an NSS order (see MacHale-Manning, 2016, Theorem 3, page 2); hence A340511 is a subsequence of A066085.

However, there exist infinitely many NSS orders that are not NCLT orders (see MacHale-Manning, 2016, Corollary 19, page 6) and these NSS-CLT orders are listed in this sequence.

Theorem: The number 224*p is an NSS-CLT order for all primes p <> 2, 3, 5, 7, 31 (see MacHale-Manning, 2016, Theorem 18, page 6). So, 10528, 11872, 13216, 13664, 15008, 15904, ... are other terms.

Links

Table of n, a(n) for n=1..12.

Des MacHale and J. Manning, Converse Lagrange Theorem Orders and Supersolvable Orders, Journal of Integer Sequences, 2016, Vol. 19, #16.8.7.

Example

There exist 197 groups of order 224, and one of these groups is NSS-CLT (see MacHale-Manning, 2016, Theorem 8, page 5); this group is NSS (A066085) but satisfies the converse of Lagrange theorem (CLT): for all divisors d of 224, this group has at least one subgroup of order d; hence, 224 is a term.

Crossrefs

Equals A066085 \ A340511.

Cf. A340517.

Sequence in context: A146745 A015048 A229589 * A232804 A032802 A224431

Adjacent sequences: A341045 A341046 A341047 * A341049 A341050 A341051

Keyword

nonn,more

Author

Bernard Schott, Feb 04 2021

Status

approved