Frattini Subgroup Research Articles

Notes on Splitting Extensions of Groups

In [1] Gaschütz has shown that a finite group G splits over an abelian normal subgroup N if its Frattini subgroup ϕ(G) intersects N trivially. When N is a non-abelian nilpotent normal subgroup of G the condition ϕ(G)∩ N = 1 cannot be satisfied: for if N is non-abelian then the commutator subgroup C(N) of N is non-trivial. Now N is nilpotent, whence 1 ≠ C(N)⊂ϕ(N). Since G is a finite group, therefore, by (3, theorem 7.3.17) ϕ⊂ϕ(G). It follows that ϕ(G) ∩ N ≠ 1. Thus the condition ϕ(G) ∩ N = 1 must be modified. In §1 we shall derive some similar type of conditions for G to split over N when the restriction of N being an abelian normal subgroup is removed. In § 2 we shall give a characterization of splitting extensions of N in which every subgroup splits over its intersection with N.

Frattini Subgroups in Finitely Generated Linear Groups

Frattini Subgroups in Finitely Generated Linear Groups

A note on the automorphism group of a 𝑝-group

The relation between the order of a p-group and its automorphism group has been the subject of several papers, see [1], [2], and [4]. The existence of outer-automorphisms of a finite p-group was proved by Gaschiitz [3], but the question of the size of the automorphism group of a p-group still remains. In this paper we will prove that the order of the automorphism group of a finite nonabelian nilpotent class two group is divisible by the order of the group. It should be noted that the above result is stated in [4], but the proof is invalid; see [2]. In this paper G will denote a finite nonabelian nilpotent class two p-group. Z, G', 1 and A (G) will denote the center, derived subgroup, Frattini subgroup and automorphism group of G. S will denote a set of elements { a, b, , f } CG such that G/G'= (a G') X X(f*G'). Let ka>-kb > * * kf and k', , kJ be the orders of a, b, * * *, f modulo G' and Z respectively. If r is a rational number then [ [r] ] = max { 1, r }.

Generalized Frattini subgroups of finite groups

The purpose of this paper is to generalize some of the fundamental properties of the Frattini subgroup of a finite group. For this purpose we call a proper normal subgroup H of G a generalized Frattini subgroup if and only if G = NG(P) for each normal subgroup L of G and each Sylow p-subgroup P, p is a prime, of L such that G = HNG(P). Here NG(P) is the normalizer of P in G. Among the generalized Frattini subgroups of a finite nonnilpotent group G are the center, the Frattini subgroup, and the intersection L(G) of all selfnormalizing maximal subgroups of G. The product of two generalized Frattini subgroups of a group G need not be a generalized Frattini subgroup, hence G may not have a unique maximal generalized Frattini subgroup. Let H be a generalized Frattini subgroup of G and let K be normal in G. If K/H is nilpotent, then K is nilpotent. Similarly, if the hypercommutator of K is contained in H, then K is nilpotent. We consider the Fitting subgroup FίG) of a nonnilpotent group G, and prove F(G) is a generalized Frattini subgroup of G if and only if every solvable normal subgroup of G is nilpotent. Now let H be a maximal generalized Frattini subgroup of a finite nonnilpotent group G. Following Bechtell we introduce the concept of an iϊ-series for G and prove that if G possesses an iJ-series, then H = L(G).

On the generalized frattini subgroup of a finite group

On the generalized frattini subgroup of a finite group

The automorphism group of finite p-groups

It is well known [2; Theorem 12.2.21 that if G is a finite p-group with Frattini subgroup @, then the group 5 of automorphisms of G fixing G/Q elementwise is a finite p-group and therefore nilpotent. In this paper we shall obtain an upper bound for the nilpotency class of 3 (Theorem 3). This is done by relating a certain central series of G to a central series of 5. The argument uses induction (Theorem 2 giving information about the center of 3) and depends on a partly well-known general result (Theorem 1 and Corollary 1) showing how certain subgroups of a group lead to normal subgroups of its group of automorphisms. We conclude by applying the theory to Abelian p-groups, wreath products, etc.

Frattini subgroups and Φ-central groups

Frattini subgroups and Φ-central groups

A condition for a finite group to be nilpotent

Let xe be a class of groups such that: (i) If G is in 3C, then every homomorphic image of G is in 3C. (ii) If G is finite and G/4(G) is in 3C, where +(G) is the Frattini subgroup of G, then G is in XC. Examples of such classes are the class of nilpotent groups and the class of supersolvable groups. Others can be found in a paper by Baer [1]. In this note a theorem of P. Hall on nilpotent groups is proved as a corollary to the following:

A characterization of the Jacobson radical

In this little note we give a new characterization of the Jacobson radical of an associative ring, one which is analogous to a standard description of the Frattini subgroup of a group (see e.g. [2, p. 156]). Our characterization is a generalization to the case of arbitrary rings of an observation of L. Fuchs [1 ] on rings with a unit element. Let R be an associative ring (in the sequel briefly ring). An element a of R is said to be left quasi-regular if there exists in R an element b such that a +b +ba = 0. The set of all elements x(E R) which are such that rx is left quasi-regular for every r(CER) is a (two-sided) ideal in R. This ideal is denoted by JI(R) .The corresponding "right" concepts right quasi-regular and Jr(R) are defined in an analogous way. It is well known that J1(R) = fJ(R) for every ring R (see [3]).1 This ideal is called the Jacobson radical of the ring R and it is denoted by J(R). Let A be a subset of the ring R. We denote by { A } , the left ideal of R generated by the set A. If {A } I=R, then we say that A is a left generating system of the ring R. (In this case A generates R as a left R-module.) Consider the set q5(R) of all those elements x of R, for which, with every s(ECR), xs can be cancelled from any left generating system of R; in other words: for which, whenever R { A, xs } for a subset A of R, then also R= {A } 1. The corresponding "right" concepts { A },r and 4,(R) are defined in an analogous way. We prove the following result.

Remarks on a paper of Hobby and Wright

C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

The Frattini Subgroups of Finitely Generated Groups

Proceedings of the London Mathematical SocietyVolume s3-11, Issue 1 p. 327-352 Articles The Frattini Subgroups of Finitely Generated Groups P. Hall, P. Hall King's College, CambridgeSearch for more papers by this author P. Hall, P. Hall King's College, CambridgeSearch for more papers by this author First published: 1961 https://doi.org/10.1112/plms/s3-11.1.327Citations: 62AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-11, Issue11961Pages 327-352 RelatedInformation

The Frattini subgroup of ap-group

The Frattini subgroup of a<i>p</i>-group

Poznámka k jednomu problému týkajícímu se Frattiniho podgrup

Poznámka k jednomu problému týkajícímu se Frattiniho podgrup

The Frattini subgroups of abelian groups

The Frattini subgroups of abelian groups

The Frattini subgroup of a direct product of groups

The Frattini subgroup of a direct product of groups

Associative regular operations on groups corresponding to a property 𝒫

In previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element. Golovin [ 3 ] showed how to obtain all operations satisfying the last three conditions and called such operations regular multiplications. Below we construct for every property (d, applicable to groups, a corresponding regular multiplication. If (d persists homomorphically and ceases freely, we shall show that this multiplication is associative. (d is said to persist homomorphically if from the fact that a group satisfies (P it follows that every homomorphic image of the group also satisfies (Y. On the other hand, (P is said to cease freely if for every set of groups satisfying (P their free product does not satisfy (P. We introduce a number of preliminary concepts which are all contained in Moran [6]. N(X) is said to be a normal subgroup function if it associates with every group G a normal subgroup N(G) of G. However, we are interested in the following more restrictive class of normal subgroup functions. N(X) is said to be a strongly characteristic subgroup function if N(X)o ? N(Xq) for every homomorphism 4 of X. This is equivalent to the condition that the normal subgroup be prescribed in such a manner that when extra defining relations are imposed on the group then the elements which originally belonged to the subgroup continue to remain therein. It seems that most of the well known normal subgroup functions, such as the Frattini subgroup, are of this type. Let F denote the free product of an arbitrary set of groups Ga, aECM, and N(X) be a normal subgroup function, then corresponding to N(X) we have the normal subgroup [N(Ga), G# ]F of F. [N(Ga), G# ]F is the normal subgroup of F generated by all commutators of the form [ga, g#], where gaCN(Ga), gEzG# and a and d are distinct elements of the index set M. We now define the N*-muitiplication of the groups Ga, ca EM, byl

On prime-power groups with two generators

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be writtenFor any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.