Finite groups with a unique subgroup of order p
نویسنده
چکیده
This is a toy example to illustrate some of the techniques of fusion and transfer as described in Gorenstein’s text Finite Groups. Finite groups with a unique subgroup of order p (p an odd prime) are classified in terms of a cyclic p′-extension of relatively simple combinatorial data. This note is intended to address a slightly misstated exercise in somewhat more detail and to be a toy example of the theory of fusion. Likely this classification was still child’s play in the 1930s, but it does give a clean, explicit description of a reasonably natural sounding class. The original misstated exercise was to show that if a finite group had a unique subgroup of order p, for some prime p, then that subgroup was central. As the non-abelian group of order 6 and p = 3 shows, this is an absurd claim. However, the groups that do occur break down naturally into the central case and a cyclic extension. The central case is elegantly described by Burnside’s N/C theorem. Notation and concepts are as in [1], especially chapters 5 and 7. Definition 1. A finite group is called p-nilpotent if it has a normal subgroup of order coprime to p and index a power of p (p some prime). A finite group that is not p-nilpotent is called p-length 1 if it has a normal p-nilpotent subgroup of index coprime to p. We will “p-nilpotent or p-length 1” to just “p-length at most 1”. The name pnilpotent comes from several important similarities of p-nilpotent groups to nilpotent groups, the easiest to describe is simply that a finite group is nilpotent if and only if it is p-nilpotent for all primes p. Another way to describe p-nilpotent groups is as those groups which have a normal p-complement. These groups have been studied for more than 100 years, and are involved in one of the earliest results in “fusion”: Lemma 2 (Burnside). If a Sylow p-subgroup is centralized by its normalizer, then the whole group is p-nilpotent. Proof. This is [1, Th. 7.4.3, p.252]. In fact modern methods (as in 1930s) have improved Burnside’s result to: Lemma 3. If a Sylow p-subgroup P of G is abelian and N = NG(P ), then P = (P ∩N ′)× (P ∩ Z(N)). Proof. This is [1, Th. 7.4.4, p.253]. We also need the well-known classification in case G is itself a p-group: 2000 Mathematics Subject Classification 20D20 (20D10) c © 2009, Jack Schmidt
منابع مشابه
Finite groups with $X$-quasipermutable subgroups of prime power order
Let $H$, $L$ and $X$ be subgroups of a finite group$G$. Then $H$ is said to be $X$-permutable with $L$ if for some$xin X$ we have $AL^{x}=L^{x}A$. We say that $H$ is emph{$X$-quasipermutable } (emph{$X_{S}$-quasipermutable}, respectively) in $G$ provided $G$ has a subgroup$B$ such that $G=N_{G}(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (with all Sylowsubgroups, respectively) $...
متن کاملON p-NILPOTENCY OF FINITE GROUPS WITH SS-NORMAL SUBGROUPS
Abstract. A subgroup H of a group G is said to be SS-embedded in G if there exists a normal subgroup T of G such that HT is subnormal in G and H T H sG , where H sG is the maximal s- permutable subgroup of G contained in H. We say that a subgroup H is an SS-normal subgroup in G if there exists a normal subgroup T of G such that G = HT and H T H SS , where H SS is an SS-embedded subgroup of ...
متن کاملON THE CHARACTERISTIC DEGREE OF FINITE GROUPS
In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. We define the characteristic degree of a subgroup H in a finite group G as the ratio of the number of all pairs (h,α) ∈ H×Aut(G) such that h^α∈H, by the order of H × Aut(G), where Aut(G) is the automorphisms group of G. This quantity measures the probability that H can be characteristic ...
متن کاملThe nc-supplemented subgroups of finite groups
A subgroup $H$ is said to be $nc$-supplemented in a group $G$ if there exists a subgroup $Kleq G$ such that $HKlhd G$ and $Hcap K$ is contained in $H_{G}$, the core of $H$ in $G$. We characterize the supersolubility of finite groups $G$ with that every maximal subgroup of the Sylow subgroups is $nc$-supplemented in $G$.
متن کاملA Simple Classification of Finite Groups of Order p2q2
Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, respectively. In this paper, we show that up to isomorphism, there are four groups of order p^2q^2 when Q and P are cyclic, three groups when Q is a cyclic and P is an elementary ablian group, p^2+3p/2+7 groups when Q is an elementary ablian group an...
متن کامل