Math, Comp & Art

at Heart

veracity (n) - Habitual truthfulness; honesty

March 25, 2020, 9:32 a.m.

Groups, Subgroups and Algebraic Structure

By Maurice Ticas

Tags:

algebra

group theory

symmetries of a hexagon

First exposure to group theory is most often from undergraduate math studies. An axiomatic approach to groups is established, and from that starting point a rich theory is developed. Well known textbooks on the subject are Joseph A. Gallian's Contemporary Abstract Algebra and Charles C. Pinter's A Book of Abstract Algebra.

Here we'll give a very high level overview of the subject to summarize the material from our Secolinsky group theory publication. Anyone wanting to better understand groups is invited to read its beginning theoretic development. The publication will help you understand the details and concrete nature of the rich subject.

Let's start with Gallian's one- and two-step subgroup tests. The one-test is done from first showing the subset of a group is nonempty. Then from two elements \(a,b\) in the subset, one must show that \(a b^{-1}\) is also in the subset. Alternatively, the two-step is done from showing that the subset is closed under the operation, and that the subset is closed under taking inverses. These one- and two- step subgroup tests are clarified here before you access the Secolinsky publication on group theory.

The mathematician Patrick Reany wrote that "... the main goal of group theory is to classify all groups and determine their structure. And part of knowing their structure is to know what subgroups exist in any given group". The symmetric group \(S_n\) on a set of n elements, Abelian groups, the group \(U(n)\) of units modulo n, normal groups, simple groups, factor/quotient groups, the standard and general linear groups, the normalizer \(N(H)\) of a subgroup \(H\), and the center \(Z(G)\) of a group are all introduced in the groupTheory.pdf publication. In the publication we establish the result that the conjugates \(xHx^{-1}\) of a subgroup \(H\) are themselves subgroups that preserve the structure of the group \(H\).

Conjugacy classes of a group element are introduced as a way to partition a group. This partition is useful for deriving a proof of The Class Equation. The notation \(G:H\) is used to denote the set of all left cosets of the subgroup \(H\) in \(G\), and its size \(|G:H|\) is referred to as its index.

(The Class Equation)
Let \(G\) be a finite group, \( \mathcal{A} \) be the collection of all conjugacy classes, and \(I\) be the set of representative elements -- one for each equivalence class in \(\mathcal{A}\). Then \(|G| = \sum_{a \in I } |G : C(a) | \) where \( C(a) \) is the centralizer of the representative group element \( a \), i.e. the set of all elements \( g \in G \) that commute with element \( a \).


It is an enumerative proof that adds the sizes of representative conjugacy classes in the group of order greater than 1 with those sizes of group elements whose conjugacy class are of size equal to 1. The number of those that fall into the latter are precisely those that belong to the center of the group. This is because an element of a group is in the group's center if and only if its conjugacy class is of size 1.

Lagrange's Theorem, along with the index of the subgroup \(C(a)\) in \(G\), starts the proof of The Class Equation. Starting this way, we then have that the order of \(G\) is \(|G:C(a)| \: |C(a)| \). Manipulating the equality then gives that \(|G|/|C(a)| = |G:C(a)| \). At this point, it's important to interpret \(|G:C(a)|\) as the number of elements in \(G\) that are conjugate to \(a\). Because these conjugacy classes partition the group \(G\), we then have the classic result in terms of the indexes of centralizers.

Lagrange's Theorem leads us to more questions. Specifically, what restrictions on the necessary condition of Lagrange's Theorem would allow us to have a partial converse. This direction of thought leads us to study Cauchy's Theorem, the Sylow Theorems, and Hall's Theorem.

Being adept at using mathematical logic is needed to consume the Secolinsky group theory publication. The publication illustrates the contrapositive of the Fundamental Theorem of Cyclic Groups as an efficient way to show that a group isn't cyclic. It gives a counterexample to the converse of the statement that \(G\) is cyclic implies \(G\) is Abelian using \(U(15)\), the group of units modulo 15.

The publicaiton limits the discussion of permuations to the finite set \([1 \ldots, n]\). Nevertheless, permutations are defined to be a bijection from any set to itself. Pinter gives a great introductory explanation in Chapters 7,8 on these concepts.

Historically, these sets of permutations were the first groups that mathematicians studied. Cayley's Theorem states that any group is isomorphic to a group of permutations. It's this theorem that connects our modern axiomatic treatment of group theory to its historical beginnings of just studying groups of permutations.

Cayley's Theorem is a statement about structure. The construction of an isomorphism takes center stage in its proof. Gallian makes note of giving the isomorphic construction a name: the left regular representation of the group. He goes on to show that the group consisting of all left regular representations of \(U(12)\) has a Cayley table that resembles the Cayley table for \(U(12)\). Remarkable indeed that he continues to share with us that the punctured complex plane with the origin taken out is isomorphic to the unit disc when considering the operation of both sets to be complex multiplication, or that the real line with the operation of addition is isomorphic to the complex plane with its standard definition of addition.

Towards the end of the group theory Secolinsky publication, an additional piece of knowledge about groups is shared. The kernel of a homomorphism \(\phi\), \(\ker \phi \), is a normal subgroup. This lastly entails in the publication that the quotient group \(\phi(G)\) / \(\ker \phi \) is isomorphic to \( \phi(G) \).

Our last question for this post is whether a group can be expressed as the union of two proper subgroups. Mira Bhirgava's "Groups as Unions of Proper Subgroups" gives a nice simple proof using the symmetric difference operation on groups to produce a counterexample.

If \(A, B\) are any two nonempty sets, then their symmetric difference is the set \( (A + B) := (A - B) \cup (B - A) \).

The paper goes on to generalize when and how to express a group into a union of \(n\) proper subgroups. The results are different when the requirement for the group covering changes to consist only of normal proper subgroups.

There are 0 comments. No more comments are allowed.