Moduli of abelian varieties and pdivisible groups chingli chai and frans oort abstract. Elements of e p are group homomorphisms from h p into itself, with ring multiplication given by composition. An elementary abelian 2 group or boolean group is a group satisfying the following equivalent conditions. We apply the result to the modular curve x0pqp to show that the unrami. Here the ps are primes and the rs are positive integers primary. Pdf on groups which are the product of abelian subgroups. Priddi let g be a finite pgroup, let eg be a free contractible gspace, and let n. Journal of algebra 102, 155161 1986 pure protective resolutions in the category of pprimary abelian groups doyle cutler department of mathematics, university of california, davis, california 95616 communicated by i.
Finitelygeneratedabeliangroups millersville university. 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. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication. 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. A finite group is a p group if and only if its order the number of its elements is a power of p. In particular, an equivalent condition will be derived for the group zn. Also called primary group a p pgroup, but no relation to ngroup. These are often called its p pprimary parts or p pprimary components. The segal conjecture for elementary abelian pgroupsii. Relation with other properties stronger properties. Fuller received september 18, 1990 the rproduct of a family gjjit, of abelian p groups is the torsion subgroup of nier gi, which we denote by ni,gi.
In mathematics, specifically in group theory, an elementary abelian group or elementary abelian p group is an abelian group in which every nontrivial element has order p. If the elementary abelian group phas order pn, then the rank of pis n. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may. An abelian group is a set, together with an operation. 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. However, every group of order p 2 is abelian the dihedral groups are both very similar to and very dissimilar. Therefore, an abelian group is a direct sum of a divisible abelian group and a socalled reduced group, i.
Fuller received september 18, 1990 the rproduct of a family gjjit, of abelian pgroups is the torsion subgroup of nier gi, which we denote by ni,gi. In this section, we also introduce the property ak. First, let abe an abelian group isomorphic to z p, where pis a prime number. Pdf characteristic properties of large subgroups in primary. In this paper, we give a complete classification of finite pgroups all of whose subgroups of index p 2 are abelian.
New zealand journal of mathematics volume 36 2007, 159161 a note on primary abelian n. Pure projective resolutions in the category of pprimary. Because a cyclic group is abelian, each of its conjugacy classes consists of a single. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. The basic subgroup of p groups is one of the most fundamental notions in the theory of abelian groups of arbitrary power. It is either the trivial group or is isomorphic to the additive group of a field of characteristic two. Primary decomposition and finitely generated abelian groups. We define the class of \p\primary abelian almost \n\\\sigma \groups and study their basic properties. It is either the trivial group or is isomorphic to the additive group of a field of characteristic two it is a group of exponent at most two all the nonidentity elements of the. Fp, the moduli space of polarized abelian varieties of dimension g in positive. To qualify as an abelian group, the set and operation. A divisible abelian group is a direct summand of each abelian group containing it.
For a prime power p k, the group zp k z is called a primary cyclic group. The basic subgroup of pgroups is one of the most fundamental notions in the theory of abelian groups of arbitrary power. Let h be a subgroup of a solvable group g, and assume that g. It is isomorphic to the additive group of a vector space over the field of two elements. Abelian p groups are also called p primary or simply primary. The elementary abelian groups are actually the groups cp. With abelian groups, additive notation is often used instead of multiplicative notation. Every ring is an abelian group with respect to its addition operation. Griffith, infinite abelian group theory, the university of chicago press, chicago, 1970. 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 zpz n for n a.
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. In sections 3, we turn to srings over elementary abelian pgroups. 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 fundamental theorem of abelian groups states that every finitely generated abelian group is a finite direct product of primary cyclic and infinite cyclic groups. Conversely, suppose that ais a simple abelian group. Proof under the hypotheses of this theorem, the results in section 2 together with theorem 3. We prove that the 2primary torsion subgroups of k 2. Much less known, however, is that there is a description of autg, the auto. The number p must be prime, and the elementary abelian groups are a particular kind of p group. Given a finite group g, the sylow theorems guarantee the existence of a subgroup of g of order p n for every prime power p n that divides the order of g. 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. If any abelian group g has order a multiple of p, then g must contain an element of order p.
Thus we should appreciate the results we have above for abelian groups. For example, if g is the additive group of integers, gxkg is the additive group of padic integers modulo the subgroup of rational integers. Primary decompositions in abelian rcategories 93 saturated multiplicative system in cif the following conditions are satis ed. Finite pgroups all of whose subgroups of index p2 are. An elementary abelian group is a group that satisfies the following equivalent conditions. 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. These are often called its p p primary parts or p p primary components. There are no nonabelian simple groups of order prmwhere pis a prime, r 1, p mand prm m proof. Herstein received january 11, 1984 let a and b be abelian groups with the property that there exists an isomorphism from. Finite pgroups all of whose subgroups of index p 2 are abelian.
Pdf elementary abelian groups of rank 5 are dcigroups. For example, if g is the additive group of integers, gxkg is the additive group of p adic integers modulo the subgroup of rational integers. Reza, bulletin of the belgian mathematical society simon stevin, 2012. In mathematics, specifically in group theory, an elementary abelian group or elementary abelian pgroup is an abelian group in which every nontrivial element has order p. This direct product decomposition is unique, up to a reordering of the factors. The fundamental theorem of finite abelian groups stats that every finite abelian group is a direct sum of its p p primary subgroups. Endomorphisms of h p in order to carry out our characterization, it will be necessary to give a description of e p endh p, the endomorphism ring of h p. Every finite abelian group g is a direct sum of primary cyclic groups. Pdf characteristic properties of large subgroups in.
The isomorphism preserves the subgroup structure, so we only need to know that z p has no proper nontrivial subgroups. Our main purpose is to give a description of the complexity of a module m over an elementary abelian pgroup in terms of the orders of the k generalized subgroups e, to be defined below for which the restriction to e, me, is projective theorem 1. In this paper, we give a complete classification of finite p groups all of whose subgroups of index p 2 are abelian. Statement from exam iii pgroups proof invariants theorem. Motivated by earlier work of talu for odd primes, we develop a general combinatorial method, for arbitrary. It is a restricted direct product of isomorphic subgroups, each being cyclic of prime order. Finite groups with psubnormal primary cyclic subgroups v. Having failed completely to describe the pgroups by class, how about trying to classify them by rank.
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. The prank of a nite group is the maximum of the ranks of all elementary abelian psubgroups. This chapter discusses the basic subgroups of pgroups. Here the primary decomposition is just the first and technically easier step, using the decomposition of the group order into prime powers for various primes. And of course the product of the powers of orders of these cyclic groups is the order of the original group. By axiom c, every group must have at least one element, namely the identity element 0. An elementary abelian 2group or boolean group is a group satisfying the following equivalent conditions. The p rank of a nite group is the maximum of the ranks of all elementary abelian p subgroups. Also called primary group a p p group, but no relation to n group. G is primary for the prime p if every element has order a power of p. For a reduced primary group g we define a trans finite series as. Since every element of ghas nite order, it makes sense to discuss the largest order mof an element of g.
If t is a denumerable p primary torsion group, there are natu. An abelian group is a group in which the law of composition is commutative, i. These two claims generalize our recent results in 6. Our main purpose is to give a description of the complexity of a module m over an elementary abelian p group in terms of the orders of the k generalized subgroups e, to be defined below for which the restriction to e, me, is projective theorem 1.
On products of primary abelian groups patrick keef whitman college, walla walla, washington 99362 communicated by kent r. A p group cannot always be decomposed into a direct sum of cyclic groups, not even under the assumption of absence of elements of infinite height. Finite groups containing certain abelian tisubgroups salarian, m. Appendix a the structure of finite abelian groups tcd maths home. It is the additive group of a vector space over a prime field. The model theory of finitely generated finitebyabelian groups oger, francis, journal of symbolic logic, 1984. Every nite abelian group is isomorphic to a direct product of cyclic groups of orders that are powers of prime numbers. Abelian groups a group is abelian if xy yx for all group elements x and y.
Jan 23, 2014 we define the class of \ p \ primary abelian almost \n\\\sigma \groups and study their basic properties. Every finite abelian group g is a direct sum of pprimary group. Essentially all you are doing is decomposing the group into a direct sum product of its sylow subgroups, as you could do more generally in a finite nilpotent group. In view of the associative law, we may write abcas abcwithout ambiguity. If t is a denumerable pprimary torsion group, there are natu. Let n pn1 1 p nk k be the order of the abelian group g, with pis distinct primes.
The number p must be prime, and the elementary abelian groups are a particular kind of pgroup. The fundamental theorem of finite abelian groups stats that every finite abelian group is a direct sum of its p pprimary subgroups. Monakhov abstract a subgroup h of a group g is called psubnormal in g whenever either h g or there is a chain of subgroups h h0. 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. Journal of algebra 102, 155161 1986 pure protective resolutions in the category of p primary abelian groups doyle cutler department of mathematics, university of california, davis, california 95616 communicated by i.
A group is a set g with an associative binary operation with identity such that every element is invertible. Groups peter danchev received september 2006 abstract. A classification of reduced abelian groups is known only in certain special cases. Assume that gis a simple, nonabelian group of such order. John sullivan, classification of finite abelian groups pdf. The basis theorem an abelian group is the direct product of cyclic p groups.
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