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

%I A341048 #35 Apr 03 2023 14:35:31
%S A341048 224,2464,2912,3159,3808,4256,5152,6318,6496,8288,9184,9632
%N A341048 Numbers m such that there is a group of order m that is not supersolvable (NSS) but "converse Lagrange theorem" (CLT).
%C A341048 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.
%C A341048 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.
%C A341048 Theorem: Every NCLT order is an NSS order (see MacHale-Manning, 2016, Theorem 3, page 2); hence A340511 is a subsequence of A066085.
%C A341048 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.
%C A341048 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.
%H A341048 Des MacHale and J. Manning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Manning/manning5.html">Converse Lagrange Theorem Orders and Supersolvable Orders</a>, Journal of Integer Sequences, 2016, Vol. 19, #16.8.7.
%e A341048 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.
%Y A341048 Equals A066085 \ A340511.
%Y A341048 Cf. A340517.
%K A341048 nonn,more
%O A341048 1,1
%A A341048 _Bernard Schott_, Feb 04 2021