The classification of minimal non-prime-power order groups

Gallian, Joseph A.; Moulton, David Petrie

doi:10.1007/bf01194842

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Cited by 5 publications

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“…In this paper all groups are considered to be finite. In [1], Gallian and Moulton gave a complete classification of minimal non-prime-power order groups. As a direct corollary of [2,Theorems 4.2 and 4.4], any group G with at most three conjugacy classes of proper subgroups of non-prime-power order is always solvable except for G ∼ = A 5 .…”

Section: Introductionmentioning

confidence: 99%

Finite groups with some nonnormal subgroups of non-prime-power order

Shi¹,

Zhang²,

Liang³

2015

Hokkaido Math. J.

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We prove that a group G with exactly three classes of nonnormal proper subgroups of the same non-prime-power order is nonsolvable if and only if G ∼ = A 5 , and a group G with exactly four classes of nonnormal proper subgroups of the same nonprime-power order is nonsolvable if and only if G ∼ = P SL(2, 7) or P SL(2, 8). Moreover, we prove that any group G with at most nine classes of nonnormal nontrivial subgroups of the same order is always solvable except for G ∼ = A 5 , P SL 2 (7) or SL 2 (5).

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Section: Introductionmentioning

confidence: 99%

Finite groups with some nonnormal subgroups of non-prime-power order

Shi¹,

Zhang²,

Liang³

2015

Hokkaido Math. J.

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show abstract

“…Note that a group of non-primepower order in which every non-trivial subgroup has prime-power order is a minimal group of non-prime-power order. In [3], Gallian and Moulton obtained a complete classification of non-abelian minimal groups of nonprime-power order. Obviously any non-abelian minimal group of non-primepower-order is solvable.…”

Section: Introductionmentioning

confidence: 99%