Introduction to group theory

This chapter develops the basics of group theory, with particular attention to the role of group actions of various kinds. The emphasis is on groups in Sections 1–3 and on group actions starting in Section 6. In between is a two-section digression that introduces rings, fields, vector spaces over general fields, and polynomial rings over commutative rings with identity. Section 1 introduces groups and a number of examples, and it establishes some easy results. Most of the examples arise either from number-theoretic settings or from geometric situations in which some auxiliary space plays a role. The direct product of two groups is discussed briefly so that it can be used in a table of some groups of low order. Section 2 defines coset spaces, normal subgroups, homomorphisms, quotient groups, and quotient mappings. Lagrange's Theorem is a simple but key result. Another simple but key result is the construction of a homomorphism with domain a quotient group $G/H$ when a given homomorphism is trivial on $H$. The section concludes with two standard isomorphism theorems. Section 3 introduces general direct products of groups and direct sums of abelian groups, together with their concrete “external” versions and their universal mapping properties. Sections 4–5 are a digression to define rings, fields, and ring homomorphisms, and to extend the theories concerning polynomials and vector spaces as presented in Chapters I–II. The immediate purpose of the digression is to make prime fields and the notion of characteristic available for the remainder of the chapter. The definitions of polynomials are extended to allow coefficients from any commutative ring with identity and to allow more than one indeterminate, and universal mapping properties for polynomial rings are proved. Sections 6–7 introduce group actions. Section 6 gives some geometric examples beyond those in Section 1, it establishes a counting formula concerning orbits and isotropy subgroups, and it develops some structure theory of groups by examining specific group actions on the group and its coset spaces. Section 7 uses a group action by automorphisms to define the semidirect product of two groups. This construction, in combination with results from Sections 5–6, allows one to form several new finite groups of interest. Section 8 defines simple groups, proves that alternating groups on five or more letters are simple, and then establishes the Jordan–Holder Theorem concerning the consecutive quotients that arise from composition series. Section 9 deals with finitely generated abelian groups. It is proved that “rank” is well defined for any finitely generated free abelian group, that a subgroup of a free abelian group of finite rank is always free abelian, and that any finitely generated abelian group is the direct sum of cyclic groups. Section 10 returns to structure theory for finite groups. It begins with the Sylow Theorems, which produce subgroups of prime-power order, and it gives two sample applications. One of these classifies the groups of order $pq$, where $p$ and $q$ are distinct primes, and the other provides the information necessary to classify the groups of order 12. Section 11 introduces the language of “categories” and “functors.” The notion of category is a precise version of what is sometimes called a “context” at points in the book before this section, and some of the “constructions” in the book are examples of “functors.” The section treats in this language the notions of “product” and “coproduct,” which are abstractions of “direct product” and “direct sum.”

This chapter develops the basics of group theory, with particular attention to the role of group actions of various kinds. The emphasis is on groups in Sections 1–3 and on group actions starting in Section 6. In between is a two-section digression that introduces rings, fields, vector spaces over general fields, and polynomial rings over commutative rings with identitySection 1 introduces groups and a number of examples, and it establishes some easy results. Most of the examples arise either from number-theoretic settings or from geometric situations in which some auxiliary space plays a role. The direct product of two groups is discussed briefly so that it can be used in a table of some groups of low order.Section 2 defines coset spaces, normal subgroups, homomorphisms, quotient groups, and quotient mappings. Lagrange’s Theorem is a simple but key result. Another simple but key result is the construction of a homomorphism with domain a quotient group G/H when a given homomorphism is trivial on H. The section concludes with two standard isomorphism theorems.Section 3 introduces general direct products of groups and direct sums of abelian groups, together with their concrete “external” versions and their universal mapping properties.Sections 4–5 are a digression to define rings, fields, and ring homomorphisms, and to extend the theories concerning polynomials and vector spaces as presented in Chapters I–II. The immediate purpose of the digression is to make prime fields and the notion of characteristic available for the remainder of the chapter. The definitions of polynomials are extended to allow coefficients from any commutative ring with identity and to allow more than one indeterminate, and universal mapping properties for polynomial rings are proved.Sections 6–7 introduce group actions. Section 6 gives some geometric examples beyond those in Section 1, it establishes a counting formula concerning orbits and isotropy subgroups, and it develops some structure theory of groups by examining specific group actions on the group and its coset spaces. Section 7 uses a group action by automorphisms to define the semidirect product of two groups. This construction, in combination with results from Sections 5–6, allows one to form several new finite groups of interest.Section 8 defines simple groups, proves that alternating groups on five or more letters are simple, and then establishes the Jordan-Hölder Theorem concerning the consecutive quotients that arise from composition series.Section 9 deals with finitely generated abelian groups. It is proved that “rank” is well defined for any finitely generated free abelian group, that a subgroup of a free abelian group of finite rank is always free abelian, and that any finitely generated abelian group is the direct sum of cyclic groups.Section 10 returns to structure theory for finite groups. It begins with the Sylow Theorems, which produce subgroups of prime-power order, and it gives two sample applications. One of these classifies the groups of order pq, where p and q are distinct primes, and the other provides the information necessary to classify the groups of order 12.Section 11 introduces the language of “categories” and “functors.” The notion of category is a precise version of what is sometimes called a “context” at points in the book before this section, and some of the “constructions” in the book are examples of “functors.” The section treats in this language the notions of “product” and “coproduct,” which are abstractions of “direct product” and “direct sum.”KeywordsAbelian GroupNormal SubgroupConjugacy ClassCommutative RingLinear CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Finite Permutation Groups

The early history of Sylow's theorem is surprisingly unfamiliar. A recent bulky history of mathematics, for instance, recounts that "Having arrived at the abstract notion of a group, the mathematicians turned to proving theorems about abstract groups that were suggested by known results for concrete cases. Thus Frobenius proved Sylow's theorem for finite abstract groups."1 Now a moment's thought shows that this in itself is not much of an accomplishment : any finite group can be realized as a group of permutations (Cayley's theorem), and so Sylow's theorem for permutation groups implies the abstract result. The reader then wonders whether Frobenius somehow did not know Cayley's theorem. But a glance at the paper in question2 shows that Frobenius not only mentioned that result but had included it in one of his own earlier publications. Obviously more careful analysis is needed to grasp what FROBENIUS was actually doing. The specific purpose of this paper is precisely to analyze the various different proofs that were given for Sylow's theorem in the first fifteen years after its discovery. This is of interest in itself, since the theorem is still basic, and the major lines of argument now known all turn out to have been discovered in some form by that time. But also, by tracing this single theorem through a crucial period in the development of group theory, we will see more generally how the introduction of new ideas throws new light on the same result. Indeed, we almost have a case study in levels and use of abstraction. Some of the authors still thought of changes of variable in polynomials, others applied group-theoretic reasoning to permutation groups, and finally Frobenius himself made serious use of abstract groups. In addition to the advance, the continuity in this development will also be illuminated, since we will see techniques equivalent to the construction of coset actions and quotient groups used in proofs before these concepts were formulated explicitly.

This chapter opens Part Two, devoted to the study of transformation theory, with some additional topics in group theory that arise in musical applications. Transformation groups on finite spaces may be regarded as permutation groups; permutation groups on pitch-class space include not only the groups of transpositions and inversions but also the multiplication group, the affine group, and the symmetric group. Another musical illustration of permutations involves the rearrangement of lines in invertible counterpoint. The structure of a finite group may be represented in the form of a group table or a Cayley diagram (a kind of graph). Other concepts discussed include homomorphisms and isomorphisms of groups, direct-product groups, normal subgroups, and quotient groups. Groups underlie many examples of symmetry in music, as formalized through the study of equivalence relations, orbits, and stabilizers.

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ZA is finitely generated over Am ⋊ Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

Extension groups between simple Mackey functors

A group is finitely generated if it has a finite set of generators. Finitely generated abelian groups may be classified. By this we mean we can draw up a list (albeit infinite) of “standard” examples, no two of which are isomorphic, so that if we are presented with an arbitrary finitely generated abelian group, it is isomorphic to one on our list.

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

Chapter 3 discusses actions of semigroups, groups, topological groups, and Lie groups. Each element of a group determines a permutation on a set under a group action. For a topological group action on a topological space, this permutation is a homeomorphism and for a Lie group action on a differentiable manifold it is a diffeomorphism. Group actions are used in the proofs of the Counting Principle, Cayley’s Theorem, Cauchy’s Theorem and Sylow Theorems for finite groups. The counting principle is used to determine the structure of a finite group. These groups arise in the Sylow Theorems and in the description of finite abelian groups. The orbit spaces obtained by topological group actions, discussed in this chapter, are very important in topology and geometry. For example, n-dimensional real and complex projective spaces are obtained as orbit spaces. Finally, semigroup actions are applied to theoretical computer science yielding state machines which unify computer science with mainstream mathematics.

With the rapid development of modern mathematics, math researchers have an increasing demand to take the advantage of group theory in the latest field. The group action is one of the essential parts of the group theory. In order to have a better understand of group action, this paper will describe the insight development and logic in it step by step. For this purpose, three essential theorems, which are Cauchy theorem, Sylow theorems, and orbit-stabilizer theorem, are chosen as representative examples. The proof and applications in some fields of modern science of these three theorems are discussed. In the proof, this paper emphasizes that group action can be used conveniently to solve the problem in group theory. In the application, this paper includes as many fields as possible to attach importance to group action. This paper is expected to give the opinion of how the field of group action is established and its far-reaching influence to the modern science.

Let A be a commutative Noetherian ring which is graded by a finitely generated Abelian group G. In this article, we introduce G-graded primary submodules and G-graded P-primary submodules, and establish the uniqueness of G-graded primary decomposition. We also give a new proof on existence of G-primary decomposition.

At first sight, finitely generated abelian groups and canonical forms of matrices appear to have little in common. However, reduction to Smith normal form, named after its originator H.J.S.Smith in 18

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety [𝔑 c 1 , 𝔑 c 2 ], for all c 2 ≤ c 1 ≤ 2c 2. As a consequence, we determine necessary and sufficient conditions for such groups to be [𝔑 c 1 , 𝔑 c 2 ]-capable. We also show that if c 1 ≠ 1 ≠ c 2, then a finitely generated abelian group is [𝔑 c 1 , 𝔑 c 2 ]-capable if and only if it is capable. Finally, we show that 𝔖2-capability implies capability, but there is a capable finitely generated abelian group which is not 𝔖2-capable.

This result is among the arsenal of tools that every first year algebra student obtains. A group where the order of every element is a power of p is called a p-group; a p-Sylow subgroup of G is a p-subgroup of G of maximal order The idea of Sylow's proof, which was originally stated in terms of permutation groups, is to look at the size of the equivalence classes obtained when all p-Sylow subgroups of G are conjugated by the elements of a fixed p-Sylow subgroup of G. The existence of a p-Sylow subgroup was needed for the proof of the third Sylow theorem, although the conclusion of the theorem certainly implies that there are p-Sylow subgroups. Using the Sylow results, Frobenius, in 1895 [1], proved a generalization: The number of subgroups of G of order ps is congruent to 1 modulo p whenever 1 < s < n. Most current texts show the existence of a p-Sylow subgroup and prove Sylow's third theorem using arguments that involve a group acting on a set. This method of proof for the existence of a p-Sylow subgroup was due to Miller between 1910 and 1915 [8, 9], but, according to Jacobson [9, p. 83], was forgotten until it was rediscovered in 1959 by Wielandt [14]. Krull [7] showed how Wielandt's method could be used to obtain Frobenius' generalization, and Gallagher, in 1967 [2], simplified the argument to one that depends upon the order of G rather than the group itself. Illustrating the combinatorics of finite group actions is part of the motivation to use this method of proof both to demonstrate the existence of p-Sylow subgroups in a finite group and to determine their number. In this note we offer another method to prove these results. Our combinatorial tool will be Mobius inversion on the lattice of subgroups of a finite group. We will see that an application of this method will easily lead to Frobenius' theorem, in fact, a generalization of it. Of course, part of the reason for presenting this proof is to highlight the method.