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

Author

Bernard Schott, Feb 04 2021

Keywords

• nonn
• more

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.

Examples

			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.
		

Links

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

Crossrefs

Equals A066085 \ A340511.

Cf. A340517.