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1.
A pair (G, K) in whichG is a finite group andK a normal nontrivial proper subgroup ofG is said to be an F2-pair (a Frobenius type pair) if |C G (x)|=|C G/K (xK)| for allxG\K. A theorem of Camina asserts that in this case eitherK orG/K is ap-group or elseG is a Frobenius group with Frobenius kernelK. The structure ofG will be described here under certain assumptions on the Sylowp-subgroups ofG. This author’s research was partially supported by the Technion V.P.R. fund — E.L.J. Bishop research fund. This author’s research was partially supported by the MPI fund.  相似文献   

2.
For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp 2 possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.  相似文献   

3.
For a (finite) groupG and some prime powerp n, theH p n -subgroupH pn (G) is defined byH p n (G)=〈xεG|x pn≠1〉. A groupH≠1 is called aH p n -group, if there is a finite groupG such thatH is isomorphic toH p n (G) andH p n (G)≠G. It is known that the Fitting length of a solvableH p n -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH p n -groups having Fitting lengthn, this result is “almost” best possible.  相似文献   

4.
LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andAG is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC G(a) for 1≠aA.  相似文献   

5.
LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL 1 of ℚ(T) with Galois groupG. LetL be the subfield ofL 1 fixed byH. We make the hypothesis thatL 1 admits a quadratic extensionL 2 which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL 1 then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL 2. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL 1 admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).  相似文献   

6.
Generalized Frobenius groups   总被引:2,自引:0,他引:2  
A pair (G. K) in whichG is a finite group andKG, 1<K<G, is said to satisfy (F2) if |C G (x)|=|C G/K (xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.  相似文献   

7.
LetF be a field of characteristicp>0 and letG be an arbitrary abelian group written multiplicatively withp-basis subgroup denoted byB. The first main result of the present paper is thatB is an isomorphism invariant of theF-group algebraFG. In particular, thep-local algebraically compact groupG can be retrieved fromFG. Moreover, for the lower basis subgroupB 1 of thep-componentG p it is shown thatG p/Bl is determined byFG. Besides, ifH is (p-)high inG, thenG p/Hp andH p n[p] for ℕ0 are structure invariants forFG, andH[p] as a valued vector space is a structural invariant forN 0 G, whereN p is the simple field ofp-elements. Next, presume thatG isp-mixed with maximal divisible subgroupD. ThenD andF(G/D) are functional invariants forFG. The final major result is that the relative Ulm-Kaplansky-Mackeyp-invariants ofG with respect to the subgroupC are isomorphic invariants of the pair (FG, FC) ofF-algebras. These facts generalize and extend analogous in this aspect results due to May (1969), Berman-Mollov (1969) and Beers-Richman-Walker (1983). As a finish, some other invariants for commutative group algebras are obtained.  相似文献   

8.
On the full automorphism group of a graph   总被引:11,自引:0,他引:11  
While it is easy to characterize the graphs on which a given transitive permutation groupG acts, it is very difficult to characterize the graphsX with Aut (X)=G. We prove here that for the certain transitive permutation groups a simple necessary condition is also sufficient. As a corollary we find that, whenG is ap-group with no homomorphism ontoZ p wrZ p , almost all Cayley graphs ofG have automorphism group isomorphic toG.  相似文献   

9.
Letp be a prime number,G a pro-p group, andH a closed (topologically) finitely generated subgroup ofG. We give conditions under whichH is virtually a free factor ofG, i.e., that there exists an open subgroupU ofG such thatU is the free pro-p product ofH and some other subgroup ofU. We prove that this happens if eitherG is a free pro-p group of any rank, or ifG is a free pro-p product of finitely generated pro-p groups. Research supported in part by grants from NSERC (Canada) and DGICYT (Spain).  相似文献   

10.
Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.  相似文献   

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