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A340511
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Numbers k such that there exists a group of order k which has no subgroup of order d, for some divisor d of k.
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5
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12, 24, 36, 48, 56, 60, 72, 75, 80, 84, 96, 108, 112, 120, 132, 144, 150, 156, 160, 168, 180, 192, 196, 200, 204, 216, 225, 228, 240, 252, 264, 276, 280, 288, 294, 300, 312, 320, 324, 336, 348, 351, 360, 363, 372, 375, 384, 392, 396, 400, 405, 408, 420
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OFFSET
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1,1
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COMMENTS
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Suggested by the fact that the converse to Lagrange's theorem does not hold. These numbers might be called "Non-Converse Lagrange Theorem Orders".
A subsequence of A066085. The first difference between them is that 224 is missing from the present sequence (see MacHale-Manning, 2016). The sequence of terms of A066085 not in the present sequence is infinite, and begins 224, 2464, ... [This sequence is now A341048. - Bernard Schott, Feb 15 2021]
If k is a term, gcd(k,m) = 1, then k*m is again a term. Proof: If G is a group of order k without a subgroup of order k', then G X C_m has no subgroup of order k'*m. Suppose that it has, let G' be that subgroup. For every (a,b) in G', let m_0 be a multiple of m congruent to 1 modulo k, then (a,b)^(m_0) = (a,1) in G'; let k_0 be a multiple of k congruent to 1 modulo m, then (a,b)^(k_0) = (1,b) in G'. This shows that G' itself is of the form H X C_{m'}, where H is a subgroup of G and m' divides m. We have |H|*m' = k'*m, so |H| = k' and m' = m, contradicting with our assumption that G has no subgroup of order k'.
On the other hand, if gcd(k,m) > 1, then k*m need not be a term, as 56 is here but 224 is missing. In fact, N has a proper divisor here but N itself is not in this sequence if and only if N is in A341048. For the "only if" part, if N = k*m is a CLT order and k is a NCLT order, then k is a NSS order. Since every multiple of a NSS order is a NSS order, N is a NSS order, so by definition N is in A341048. The "if" part follows from MacHale-Manning, 2016, Corollary 13, Page 5.
Conjecture: If k = p^a*q^b, where p, q are primes, q !== 1 (mod p), b >= ord(q,p), then k is a term of this sequence, unless k is an NSS-CLT order of the form described in MacHale-Manning, 2016, Theorem 8, Page 5. Here ord(q,p) is the multiplicative order of q modulo p. Moreover, if k satisfies this condition, it seems that for each NCLT group of order k, the missing orders of subgroups are of the form p^a'*q^b' where either a' = a or b' = b, and a' = a if p == 1 (mod q) or a < ord(p,q). (End)
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LINKS
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EXAMPLE
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12 belongs to this sequence because there is a group of order 12 (A_4) which has no subgroup of order 6, despite the fact that 6 divides 12.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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