Theorie der endlichen Gruppen

Kurzweil, Hans; Stellmacher, Bernd

doi:10.1007/978-3-642-58816-7

Cited by 25 publications

(48 citation statements)

References 20 publications

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“…For the definition of the basic group theoretic terminology and concepts used in this paper, such as the centralizer C G ðxÞ of an element x in a group G, the centralizer C G ðUÞ of a subgroup U in G, the center ZðGÞ of G, or the commutator subgroup G 0 of G, the reader may consult any standard book on group theory, for example [9] or [13].…”

Section: Introductionmentioning

confidence: 99%

A classification of all symmetric block designs of order nine with an automorphism of order six

Crnković

Rukavina

Schmidt

2005

J of Combinatorial Designs

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Abstract:We complete the classification of all symmetric designs of order nine admitting an automorphism of order six. As a matter of fact, the classification for the parameters (35,17,8), (56,11,2), and (91,10,1) had already been done, and in this paper we present the results for the parameters (36,15,6), (40,13,4), and (45,12,3). We also provide information about the order and the structure of the full automorphism groups of the constructed designs. # 2005 Wiley Periodicals, Inc.

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Section: Introductionmentioning

confidence: 99%

A classification of all symmetric block designs of order nine with an automorphism of order six

Crnković

Rukavina

Schmidt

2005

J of Combinatorial Designs

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“…Then we may identify j l = α (l l+1) with (l l+1) (7 8) because there is just one embedding S 6 → A 8 and α is faithful. Thus (11) may be read in A 8 as (l−1 l) (7 8) = f −1 1 (l l+1) (7 8)f 1 and we see that the order of f 1 should be 6, a contradiction to (10). Summing up, we have …”

Section: Invariant Transversal Subspacesmentioning

confidence: 85%

Imprimitive groups highly transitive on blocks

Bartolone

Musumeci²,

Strambach³

2004

Journal of Group Theory

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We classify imprimitive groups acting highly transitively on blocks and satisfying conditions common in geometry. They can be realized as suitable subgroups of twisted wreath products.In contrast to the theory of primitive permutation groups, the literature for imprimitive groups appears needy. The best known construction principle for imprimitive groups is given by wreath products of two groups U andḠ.Suppose that U is a vector space and let V be the direct sum t i=1 U i of t copies of U . IfḠ is a subgroup of the symmetric group S t acting transitively on the set {U 1 , . . . , U t } of components of V and α :Ḡ U t → GL(U ) is a linear representation of the stabilizer inḠ of the component U t , then the twisted wreath product G = U wr αḠ may be seen as a group of affine mappings acting transitively on a suitable system of affine subspaces of V and having t blocks there. The abelian group T consisting of all translations is the corresponding inertial group, i.e. the normal subgroup of G leaving each block fixed. Restricting to aḠ-invariant subspace W of V , for which there is an integer m ≤ t such that the projection W −→ i m r=i 1 U r is an isomorphism with respect to any m-subset of {1, . . . , t}, we obtain a subgroup of G having the same system of blocks as G and inertial group given by the translations corresponding to W . In particular, the inertial group induces a regular action on each block and acts sharply transitively on m-tuples of independent points, i.e. points no two of which lying in the same block. If the representation α is transitive, then U is finite and the stabilizer of a block is 2-transitive on it. * Research supported by M.U.R.S.T..

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“…Thus G is a p-group. If every two nontrivial subgroups of |G| have nontrivial intersection then either G is cyclic or p = 2 and G is a generalized quaternion group (see for instance [8], Satz 5.3.7). Suppose G has two subgroups a 1 and a 2 of order p m and p k respectively.…”

Section: Proposition 55 If G Is a Finite Group Then K(g) = |G| If Anmentioning

confidence: 99%

“…For instance the maximal order of elements of A 7 is 7, but the normalizer of a subgroup of order 7 in A 7 , has order 21 (there is not a subgroup of order 14) and it is easily seen that k( A 7 ) = (7!/2)/(2 · 6) = 210 as A 7 contains a dihedral subgroup of order 12. The maximal order of elements of A 8 is 15, that is the same as for S 8 , but there is not a dihedral subgroup of order ≥ 15. Hence k(A 8 ) = k(S 8 ).…”

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confidence: 99%

On symmetric representations of groups of automorphism of bordered Klein surfaces

Bagiński

Gromadzki

2012

RACSAM

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An automorphism of a bordered compact Klein surface induces a permutation of its boundary components and we study the corresponding representations of groups of automorphisms of such surfaces in the corresponding finite symmetric groups. The principal result gives a characterization of those abstract representations of a finite group G in symmetric groups which, up to the natural equivalence, proceed from compact Klein surfaces on which G acts as a group of dianalytic automorphisms. We deal with such representations for actions of G given by a so called smooth epimorphisms → G, where are so called nonEuclidean crystallographic groups (NEC-groups). For such data we calculate the kernels of our representations which allow, as an application, to get some results on the minimal degree of such representations for finite abstract groups.

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Theorie der endlichen Gruppen

Cited by 25 publications

References 20 publications

A classification of all symmetric block designs of order nine with an automorphism of order six

A classification of all symmetric block designs of order nine with an automorphism of order six

Imprimitive groups highly transitive on blocks

On symmetric representations of groups of automorphism of bordered Klein surfaces

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