Abelian groups generalise the arithmetic o addeetion o integers. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. A finite p group is a finite group whose order is a power of the prime, p. Abelian subgroups of pgroups mathematics stack exchange. Notation and terminology that will arise are briefly described below. This simpli es many things compared to general varieties, but it also means that one can ask harder questions. Finite non abelian and every proper subgroup is abelian implies metabelian. To qualify as an abelian group, the set and operation, a, must satisfy five requirements known as the abelian group axioms. Let p be a prime, and let g be a pgroup of maximal class of order pn, which is metabelian. Some older content on the wiki uses capital a for abelian. Agam shah, pcworld, china adds a quantum computer to highperformance computing arsenal, 4 may 2017.
Reza, bulletin of the belgian mathematical society simon stevin, 2012. However, most pgroups are of class 2, in the sense that as n. The symbol is a general placeholder for a concretely given operation. Finite pgroups in representation theory 3 iii the rank of an elementary abelian pgroup e is the integer ranke given by e pranke. Pseudofree families of finite computational elementary. That is, ranke is the dimension of e viewed as f pvector space.
Finite nonabelian and every proper subgroup is abelian. This is the hypothesis that gross and berger assumed in 6 and 1. G is a pgroup if i g is a torsionfree metabelian group. An abelian p group is homocyclic of type pe if e i e, for all i 1n. There are many common situations in which p p p groups are important. The atmodules possess a number of other nice properties 22, which will be proved in a subsequent paper.
If any abelian group g has order a multiple of p, then g must contain an element of order p. The following is reworded for further clarity for the naysayers out there. Let g be a nonabelian finite pgroup with an abelian maximal subgroup m and cyclic center. Throughout the following, g is a reduced pprimary abelian group, p 5, and v is the group of all automorphisms of g. A finite group is p abelian if and only if it is a section of a direct product of an abelian pgroup and a group of exponent p. Thus there is an easy characterization of pgroups of class 1. Fp, the moduli space of polarized abelian varieties of dimension g in positive. And of course the product of the powers of orders of these cyclic groups is the order of the original group. You can get hold of this paper if you have access to springers collection. A complex abelian variety is a smooth projective variety which happens to be a complex torus. Among the points p on x are the socalled nonsingular ones.
Notice, however, that there is a largest possible class for each order. Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. In particular, the sylow subgroups of any finite group are p p p groups. Then there is a positive integernn,k such that whenp nn,k,l k n p has the strong sperner property. A pgroup cannot always be decomposed into a direct sum of cyclic groups, not even under the assumption of absence of elements of infinite height. Thesearethenotesformath731,taughtattheuniversityofmichigan infall1991,somewhatrevisedfromthosehandedoutduringthecourse. Now let us restrict our attention to finite abelian groups.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Suppose is a non abelian finite group such that every proper subgroup of is an abelian group. By problem 2 of homework 35 p is abelian so z p p, a contradiction which shows that the former case is not possible. Large abelian subgroups of finite pgroups george glauberman august 19, 1997 1 introduction let pbe a prime andsbe a nite pgroup. For the main result of the papers by cepulic and pylavska, and zhang, an and xu, classifying the p groups all of whose proper nonabelian subgroups are metacyclic, we offer a proof which is shorter and not so involved. An elementary abelian group is a group that satisfies the following equivalent conditions. Equivalently, a group g is metabelian if and only if there is an abelian normal subgroup a such that the quotient group ga is abelian subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms metabelian groups are solvable.
Another aspect of pgroup theory is the enumeration of. The group f will serve as a frame for our discussion for if g is any metabelian twogenerator group of exponent p. Let g be a nonabelian finite p group with an abelian maximal subgroup m and cyclic center. Also, for p 2 the class pn,p contains the elementary abelian group of order pn and, for every prime divisor q of p.
For metabelian pgroups in gt2 with p 2 we can show that such groups are nilpotent of class exactly 3 theorem 3. It is a restricted direct product of isomorphic subgroups, each being cyclic of prime order. A nonabelian group g is said to be metabelian if g is abelian. We can express any finite abelian group as a finite direct product of cyclic groups. Now, for an abelian pgroup to have finite rank is equivalent to satisfying the.
Every elementary abelian p group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. The result is clear if jgjis a prime power in particular, if jgjis prime. If k 1, 2, 3, this property is subgroup inherited in the sense that if k is the bound on the number of generators of all abelian normal subgroups. The genus spectrum of a finite group g is the set of all g such that g acts faithfully on a compact riemann surface of genus g. This strengthens our recent result published in comment.
There is also a theorem on the order of an abelian subgroup that is always contained in a metabelian group of order pm, where p is a prime. Hence g is isomorphic to a subgroup of gl p, f, where f is the complex numbers, for instance. A pgroup g is said to beirregular if it is not regular. We will see that every basis of a free abelian group is of the same cardinality theorem ii. We know that groups of prime order p are cyclic, so p z p is cyclic. Throughout this thesis a p group will mean a finite p group. The finite heisenberg group h3,p of order p3 is metabelian. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a. Abelian subgroups of pgroups simon fraser university. The group has a minimal normal subgroup, and by 1 this subgroup is a p group for some prime p. We prove this by induction on the power m of the order pm of the p group.
Then show that there exists c in g such that the order of c is the least common multiple of the orders of a, b. B er k ovic 3 gives an example of a pgroup with exactly 5p elementary abelian subgroups of order p7. Therefore g contains exactly p2 subgroups of maximal class and index p. Primary 11m41, 05a15, 15b36, secondary 20k01, 20f69, 11b36. Finite pgroups all of whose nonabelian proper subgroups. For example, a product such as \a3 b5 a7\ in an abelian group could always be simplified in this case, to \a4 b5\.
Abelian varieties are indeed abelian groups unlike elliptic curves which arent ellipses, however the use. A p group is a group in which every element has order equal to a power of p. Abstract algebra 1 definition of an abelian group youtube. Abelian quasinormal subgroups of finite pgroups sciencedirect. In group theory, an elementary abelian group or elementary abelian pgroup is an abelian group in which every nontrivial element has order p. Introduction this note supplements papers b5 and bj2. Finite pgroups with a minimal nonabelian subgroup of index. Every pgroup is periodic since by definition every element has finite order if p is prime and g is a group of order p k, then g has a normal subgroup of order p m for every 1. Berkovic uses this example to construct a group g of order 511 with exactly 25 abelian subgroups of order 5 index 54 with all of them nonnormal in g. The group gis said to be abelian if ab bafor all a,b. Motivated by earlier work of talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a. It is wellknown that the class pn,2 consists only of the elementary abelian group of order 2n. On subgroups of free burnside groups of large odd exponent ivanov, s. The basic subgroup of pgroups is one of the most fundamental notions in the theory of abelian groups of arbitrary power.
Bruner erich ossa 1 computed the complex connective ktheory of elementary abelian groups by exploiting the idempotence of h bz2 as a module over the exterior algebra eq 0. Large abelian unipotent subgroups of finite chevalley groups. Pdf explicit expressions for the transfers v i from a metabelian pgroup g of coclass ccg1 to its maximal normal subgroups m 1. For example, the additive group z is a free abelian group of rank. In mathematics, specifically group theory, given a prime number p, a p group is a group in which the order of every element is a power of p. The finite heisenberg group h 3, p of order p 3 is metabelian. In abstract algebra, an abelian group, an aa cried a commutative group, is a group in which the result o applyin the group operation tae twa group elements disna depend on the order in which thay are written.
The following commutator formulae are useful in this paper, and we will use it. So a basis of a group seems to be almost identical to a. The fundamental theorem of finite abelian groups every nite abelian group is isomorphic to a direct product of cyclic groups of prime power order. Classification of finite nonabelian groups in which every. As g has cyclic center, it has faithful irreducible representations. In what follows let r, s denote any nonempty subsets and h any subgroup of a group g. If p is the exponent of g then this is just the classical result that the maxi mal normal abelian subgroups of groups are selfcentralizing. If an abelian group gis homocyclic of type p, then gis called elementary abelian. Torsion subgroup of an abelian group, quotient is a torsion. The number p must be prime, and the elementary abelian groups are a particular kind of pgroup. In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Every nite abelian group is isomorphic to a direct product of cyclic groups of orders that are powers of prime numbers. We prove that a torsion subgroup is in fact a subgroup of an abelian group and the quotient by the torsion subgroup is a torsionfree abelian group.
This follows by induction, using cauchys theorem and the correspondence theorem for groups. If it is an abelian group, the problem reduces to classifying all groups of nilpotence class two in which every proper subgroup is abelian. Metabelian pgroups and coclass theory sciencedirect. The p groups all of whose nonabelian maximal subgroups are either absolutely regular or of maximal class, are classified. On the other hand, if g p n with n 3, then there are always non abelian groups as well. The existence of an element in an abelian group of order the. Slender nilpotent and every proper subgroup is abelian implies frattiniin. Statement from exam iii pgroups proof invariants theorem. The exponent of a group gwhich we denote by expg, is the least common multiple of the orders of all elements. A3 for any a 2a, there exists b 2a such that a b e. Using additive notation, we can rewrite the axioms for an abelian group in a way that points out the similarities with. If g is a free abelian group then the rank of g is the cardinality of a basis of g. Letn andk be arbitrary positive integers, p a prime number and lk n p the subgroup lattice of the abelianp group z p k n.
This is a set of notes for a course we gave in the second week of august in the 2006 cmi summer school at go. Formula for the number of subgroups of a finite abelian group of rank two is already determined. This chapter discusses the basic subgroups of pgroups. Large abelian subgroups of finite p groups george glauberman august 19, 1997 1 introduction let pbe a prime andsbe a nite p group. A2 there is an element e 2a such that a e a for all a 2a.
It is shown that every noncentral normal subgroup of t contains a noncentral elementary abelian normal psubgroup of t of rank at least 2. Pdf on abelian subgroups of finitely generated metabelian. In symbols, a group is termed abelian if for any elements and in, here denotes the product of and in. In fact, the claim is true if k 1 because any group of prime order is a cyclic group, and in this case any nonidentity element will. As g has an abelian maximal subgroup, these must have degree p.
We show that the coclass tree associated with the metabelian pgroups of a fixed coclass is virtually periodic. Centralizers of abelian normal subgroups of pgroups. Related facts further facts about every proper subgroup being abelian. Aug 03, 2009 it is easy to show that this implies that g is abelian. A finite non abelian group in which every proper subgroup is abelian. An abelian group is a group where any two elements commute. Splitting the automorphism group of an abelian pgroup. Strong sperner property of the subgroup lattice of an abelian. Let ff be the smallest function such that every finite p group, all of whose abelian subgroups are generated by at most n elements all of whose abelian subgroups have orders at most pn, has at most fn generators has order not exceeding pfn. That is, for each element g of a p group g, there exists a nonnegative integer n such that the product of p n copies of g, and not fewer, is equal to the identity element. Recent examples on the web microsoft is trying to chase a new quantum computer based on a new topography and a yetundiscovered particle called non abelian anyons.
The same is true for any heisenberg group defined over a ring group of uppertriangular 3. A pgroup is an sgroup if it is isomorphic to the torsion sub. With abelian groups, additive notation is often used instead of multiplicative notation. It is established that the functions f and f have quadratic order of growth. Degenerate abelian functions are distinguished by having infinitely small. Pdf quadratic form of subgroups of a finite abelian p. In 1964, john thompson introduced t2 a subgroup similar to js. An abelian group is a set, a, together with an operation that combines any two elements a and b to form another element denoted a b. On abelian subgroups of finitely generated metabelian groups. Abelian groups are generally simpler to analyze than nonabelian groups are, as many objects of interest for a given group simplify to special cases when the group is abelian. The translations of the plane form an abelian normal subgroup of the group, and the corresponding quotient is the circle group. Let pbe a prime dividing jgjand n be a minimal normal. Here we concentrate on abelian quasinormal subgroups a of a finite pgroup g. Abelian p group corresponding to a p primary part of g is the direct product of cyclic groups.
On abelian subgroups of pgroups 263 divisible by p2. The fundamental theorem implies that every nite abelian group can be written up to isomorphism in the form z p 1 1 z p 2 2 z n n. Alsosinceaqngif and only if aqnax for all cyclic subgroups x of g, we consider groups g ax with x cyclic. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian pgroups. Finite groups containing certain abelian tisubgroups salarian, m. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form z p z n. Download fulltext pdf on abelian subgroups of finitely generated metabelian groups article pdf available in journal of group theory 16. That is, thir are the groups that obey the axiom o commutativity. Pdf splitting the automorphism group of an abelian p. E is a subgroup of a p group g, maximal subject to being normal abelian and of exponent p, then any element of order at most p which centralizes e lies in e, unless perhaps p 2 and n 1. We can also tackle the case of the frobenius group. Since q p, one of these subgroups, say h,isainvariant.
For example, the conjugacy classes of an abelian group consist of singleton sets sets containing one element, and every subgroup of an abelian group is normal. Minimal nonabelian and maximal subgroups of a finite pgroup. It is the additive group of a vector space over a prime field. A finite group is a p p p group if and only if its order is a power of p. A group is metabelian if its commutator subgroup is abelian. The number of generators and orders of abelian subgroups of. Since a pgroup of maximal class is generated by two elements, the minimal number of generators of g is 3, and so the number of maximal subgroups in g is 1 q p q p2 2 p2. A group g is said to be an angroup, if all its subgroups of index pn are abelian but it contains a nonabelian. A p group gis elementary abelian if and only if gis abelian and has exponent p.
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