Automorphisms of Finite Abelian Groups

Bidwell, J. N. S.; Curran, Michael J.

doi:10.3318/pria.2010.110.1.57

Cited by 9 publications

(11 citation statements)

References 2 publications

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“…The proof is quite similar to that of Theorem 2.5 in [2] so only an outline is provided. Let F be the group generated by a, b, c, d satisfying the given relations.…”

Section: Theoremmentioning

confidence: 92%

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Corrigendum to “The automorphism group of a split metacyclic p-group”. [Arch. Math 87 (2006) 488–497]

Bidwell

Curran

2009

Arch. Math.

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We clarify the definition of the subgroup C given in the paper, correct Theorem 3.2, Lemma 4.1 and the presentation of the automorphism group of a split metacyclic p-group given in Theorem 4.2.In the paper [1] we considered the automorphism group of a nonabelian split metacyclic p-groupwhere p was odd, m ≥ 2, n ≥ 1 and 1 ≤ r ≤ min{m − 1, n}. For convenience three cases were considered:Zhou and Liu [4] have pointed out some errors in Theorem 3.2, Lemma 4.1, and Theorem 4.2 of our paper and we are grateful for their observation. The errors arose because of some poor notation associated with our automorphism c, so here we clarify its definition and correct these results. Throughout this note for group elements g and h, g h denotes h −1 gh.

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“…The proof is quite similar to that of Theorem 2.5 in [2] so only an outline is provided. Let F be the group generated by a, b, c, d satisfying the given relations.…”

Section: Theoremmentioning

confidence: 92%

“…Now by Lemma 2.1 (ii) in [2], if u = 1+ p v , for any t there is a q such that 1+u+...+u q−1 = t or such that 1+w+...…”

Section: Theoremmentioning

confidence: 99%

Corrigendum to “The automorphism group of a split metacyclic p-group”. [Arch. Math 87 (2006) 488–497]

Bidwell

Curran

2009

Arch. Math.

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“…For example, one can refer to [14] for |Aut(A)| when A is abelian or dihedral groups. Also, one can see recent two papers [1], [7] for abelian case.…”

Section: Introductionmentioning

confidence: 92%

Enumerating regular graph coverings whose covering transformation groups are ℤ_2-extensions of a cyclic group

Lee²,

Kwak³

2018

Ars Math. Contemp.

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Several types of the isomorphism classes of graph coverings have been enumerated by many authors. In 1988, Hofmeister enumerated the double covers of a graph, and this work was extended to n-fold coverings of a graph by the second and third authors. For regular coverings of a graph, their isomorphism classes were enumerated when the covering transformation group is a finite abelian or dihedral group. In this paper, we enumerate the isomorphism classes of graph coverings when the covering transformation group is a Z 2extension of a cyclic group, including generalized quaternion and semi-dihedral groups.

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“…where C U can be replaced by D U . e) Space U (see (10), (12)) and space T U T U ⊂ U (see the paragraph containing (19)) are isomorphically parametrized by Î U 1 Im and (Ì U ) Im ⊂ Î U 1 Im , respectively:…”

Section: As In the Case Ofmentioning

confidence: 99%

“…The dimension D(U) of the solution space of the above system is an upper bound on the dimension of a manifold F of the type described in the second paragraph preceding (10), because:…”

Section: Introductionmentioning

confidence: 99%

Defect and Equivalence of Unitary Matrices. The Fourier Case. Part I

Tadej

2018

Open Syst. Inf. Dyn.

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Consider the real space U of directions moving into which from a unitary N × N matrix U we do not disturb its unitarity and the moduli of its entries in the first order. dim Ê ( U ) is called the defect of U and denoted D(U ). We give an account of Alexander Karabegov's theory where U is parametrized by the imaginary subspace of the eigenspace, associated with λ = 1, of a certain unitary operator I U on Å N , and where D(U ) is the multiplicity of 1 in the spectrum of I U . This characterization allows us to establish dependence of D U (1) ⊗ . . . ⊗ U (r) on D U (k) 's, to derive formulas expressing D(F ) for a Fourier matrix F of the size being a power of a prime number, as well as to show the multiplicativity of D(F ) with respect to Kronecker factors of F if their sizes are pairwise relatively prime. Also partly due to the role of symmetries of U in the determination of the eigenspaces of I U we study the 'permute and enphase' symmetries and equivalence of Fourier matrices, associated with arbitrary finite abelian groups.

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Automorphisms of Finite Abelian Groups

Cited by 9 publications

References 2 publications

Corrigendum to “The automorphism group of a split metacyclic p-group”. [Arch. Math 87 (2006) 488–497]

Corrigendum to “The automorphism group of a split metacyclic p-group”. [Arch. Math 87 (2006) 488–497]

Enumerating regular graph coverings whose covering transformation groups are ℤ_2-extensions of a cyclic group

Defect and Equivalence of Unitary Matrices. The Fourier Case. Part I

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