How Rare Are Subgroups of Index 2?

Nganou, Jean B.

doi:10.4169/math.mag.85.3.215

Cited by 6 publications

(6 citation statements)

References 4 publications

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“…Following Nganou [14] we can apply some basic, yet very useful, group theory facts to examine the index 2 subgroups of a given group.…”

Section: Index Two Subgroupsmentioning

confidence: 99%

Characteristic subgroup lattices and Hopf–Galois structures

Kohl

2019

Int. J. Algebra Comput.

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The Hopf-Galois structures on normal extensions K/k with G = Gal(K/k) are in one-to-one correspondence with the set of regular subgroups N ≤ B = P erm(G) that are normalized by the left regular representation λ(G) ≤ B. Each such N corresponds to a Hopf algebra H N = (K[N ]) G that acts on K/k. Such regular subgroups N need not be isomorphic to G but must have the same order. One can subdivide the totality of all such N into collections R(G, [M ]) which is the set of those regular N normalized by λ(G) and isomorphic to a given abstract group M where |M | = |G|. There arises an injective correspondence between the characteristic subgroups of a given N and the set of subgroups of G stemming from the Galois correspondence between sub-Hopf algebras of H N and intermediate fields k ⊆ F ⊆ K. We utilize this correspondence to show that for certain pairings (G, [M ]), the collection R(G, [M ]) must be empty.

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“…Following Nganou [14] we can apply some basic, yet very useful, group theory facts to examine the index 2 subgroups of a given group.…”

Section: Index Two Subgroupsmentioning

confidence: 99%

Characteristic subgroup lattices and Hopf–Galois structures

Kohl

2019

Int. J. Algebra Comput.

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“…One can apply the Fundamental Theorem of Galois Theory and the results in [4] about subgroups of index 2 to obtain the following.…”

Section: Number Of Intermediate Fields Of Degreementioning

confidence: 99%

“…4.2] that C 2 is cyclic of order m/gcd(2, m); and that (G1 ⊕ G 2 ) 2 = G 2 1 ⊕ G 2 2 for every groups G 1 , G 2 [4, Theorem 4]. 1.…”

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confidence: 99%

The number of quadratic subfields

Nganou¹

2014

Irish Math. Soc. Bull.

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We use elementary field theory to compute the number of intermediate fields of degree 2 of any finite extension of fields of characteristic not 2. This leads to necessary and sufficient conditions for such extensions to have no intermediate fields of degree 2, or to have a unique intermediate field of degree 2. We obtain several applications including the number of quadratic subfields of cyclotomic extensions.2010 Mathematics Subject Classification. 06D99, 08A30. Key words and phrases. Galois group, Galois extension, finite extension, intermediate fields, quadratic closure, cyclotomic extensions, subgroup of squares.Received on 22-1-2013; revised 25-2-2014. I would like to thank the referee for his/her comments, which improved the readability of the article.

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“…Now we prove (i) and (ii). It is known (see, e.g., ) that if G is any additive group, then

2 G : = ⟨ g + g | g \in G ⟩

is a characteristic subgroup of G , the quotient

G / 2 G

is an elementary abelian 2‐group, and G has exactly

| G / 2 G | - 1

subgroups of index 2. Now set

| G | = 2^{n} d

with d odd and let S be a Sylow 2‐subgroup of G .…”

Section: Enumerating 1‐rotational Hamiltonian Cycle Systems Up To Isomentioning

confidence: 99%

“…(i) It is known (see, e.g., [35]) that if G is any additive group, then 2G := g + g | g ∈ G is a characteristic subgroup of G, the quotient G/2G is an elementary abelian 2-group, and G has exactly |G/2G| − 1 subgroups of index 2. Now set |G| = 2 n d with d odd and let S be a Sylow 2-subgroup of G. Considering that our group G is binary, then S is binary as well therefore S is either cyclic or dicyclic (see, e.g., [39], Theorem 4.4]) so that we have either |2S| = 2 n−1 or |2S| = 2 n−2 , respectively.…”

Section: Proofmentioning

confidence: 99%

Some Results on 1‐Rotational Hamiltonian Cycle Systems

Buratti

Rinaldi

Traetta

2013

J of Combinatorial Designs

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A Hamiltonian cycle system of the complete graph on v vertices (briefly, a HCS(v)) is 1-rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any integer n greatest or equal to 3, there exists a 3-perfect 1-rotational HCS(2n+1). This allows to get the existence of an infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion, a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, an imprtant lower bound for the number of non isomorphic 1-rotational (and hence symmetric) HCSs is obtained

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How Rare Are Subgroups of Index 2?

Cited by 6 publications

References 4 publications

Characteristic subgroup lattices and Hopf–Galois structures

Characteristic subgroup lattices and Hopf–Galois structures

The number of quadratic subfields

Some Results on 1‐Rotational Hamiltonian Cycle Systems

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