Chapter 8 on Abstract Groups

Abstract mathematical groups are everywhere in mathematics. A central, fundamental, mathematical abstraction, groups arise irresistibly in number theory, linear algebra, differential equations, algebraic topology, and differential geometry. Where there are symmetries, transformations, and invariants, there are mathematical groups. Where there is measurement, there are groups. Whenever one changes one’s choice of measurement units or system of coordinates, a mathematical group lurks in the background. An axis of measurement is an identified symmetry.

Groups are abstract, their study is difficult, and their key concepts require more motivation than they generally receive. Within the curriculum, groups arise, at least implicitly in treatments of linear transformations and matrices, but a more systematic study is for math majors and the important and fascinating study of group representations is a specialized topic for graduate students.

But, given an understanding of Chapter 7, the central concepts and motivation of group theory can be understood and appreciated without such specialized detail study. And so the purpose of Chapter 8 is to answer such questions as:

  • How do mathematical groups relate to measurement?
  • How do group-theoretic concepts relate to the world?
  • What kinds of questions have driven the development of group theory?
  • What motivates the transition from transformation groups to abstract groups?

Chapter 8 starts with a very simple concrete example and uses it to develop basic concepts of group theory and group representations, showing how such concepts arise even in the simplest of examples. Continuing the example as a central reference point, the chapter ends with a conceptual introduction to group representations that explains its importance and key concepts.

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2 thoughts on “Chapter 8 on Abstract Groups

  1. Here are two questions that I have about the (excellent!) chapter on abstract groups:

    1. Can you explain more why the elements of a group are like multiplication, not like addition? I can see concrete instances (such as, I don’t know how to write the T operation as an addition but do as a multiplication) but don’t know the principle. Your book several times uses multiplication in a sense that is wider than multiplication of scalar, real numbers, and I am wondering what the definition is of this broader sense.

    2. On page 480, in the third paragraph, you write “Since S3 restricts to an action on a subspace of R3, the action of S3 on R3 is reducible.” Does “restrict” here have the same meaning as “reduce”?

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    • 1. Can you explain more why the elements of a group are like multiplication, not like addition? I can see concrete instances (such as, I don’t know how to write the T operation as an addition but do as a multiplication) but don’t know the principle. Your book several times uses multiplication in a sense that is wider than multiplication of scalar, real numbers, and I am wondering what the definition is of this broader sense.

      Response:

      In Chapter 7, p 399 – 402, I discuss this issue in regard to matrices and virtually everything I say there is relevant to calling the composition of group elements “multiplication” instead of “addition.” Regarding addition, the essential point is that addition typically, if not always, refers to counting units. There may be multiple units being counted, as in a vector space or a matrix and one can add functions that share a common domain by adding their respective values at each point of the domain. Finally, as in the addition of angles, there can be a cyclic element, so that, in some contexts one would regard 360 degrees as indistinguishable from zero degrees (omitting measurements). Addition is commutative because a count does not depend upon the order of the counting. So, in all of these cases, the key formal distinction from a “multiplication” composition is that addition is always commutative.

      The term “multiplication” has a more general application, and, at least typically, relates to compositions within a system of transformations. As I put it in regards to number (p 402), “In the realm of numbers, addition relates to counting, but multiplication relates to measurement, in the sense of identifying the relationship to a unit.” But to identify a relationship to a unit is, ipso facto, to identify a relationship between two quantities. And a transformation is specified by specifying the relationship of an object being transformed to its value under the transformation. So multiplication, as applied to numbers, can be regarded as composing transformations and one generalizes this aspect of multiplication to a wider context.

      The composition of two relationships – a relationship of A to B versus B to C to find the relationship of A to C, is, thus, by very close analogy, viewed as a multiplication. And, as it turns out, compositions of transformations or of relationships are not always commutative. From this formal standpoint, as well, group composition is generically treated as a multiplication because it may not be commutative.

      Now there are groups (termed “Abelian groups” after the 19th century mathematician Abel) for which the group composition is commutative. In such cases the operation is sometimes termed addition and then one uses a plus sign to denote the composition of two group elements. From a formal perspective, this reminds one of the commutativity of the group composition, which has important implications. But it also has a broader conceptual significance in that such groups typically relate, in fact, to counting units. And there is a converse: Any system of measurements with an operation of addition (as described above) can be regarded as a group in that it obeys the group axioms.

      This last point is not only true from a formal perspective (obeying all the group axioms), but such a system can serve as invertible transformations, specifically as transformations of that very system of measurements.

      I use real numbers as the simplest example to show how this is done and what it means. So consider the real numbers. Any real number defines a transformation of the set of real numbers into itself. To wit, if “a” is a number, define Ta: x –> a + x, to be the transformation that maps any real number x to a + x. So, for example, T5(7) = 5 + 7. As an example of composition of transformations, T5 + T9 = T14 because, for any number x, (T5 + T9)(x) = T5(9 + x) = 5 + 9 + x = 14 + x = T14(x). In this example, I used the letter T to stand for “Translation.” For example T5 translates (i.e., moves) every number on the number line 5 units to the right.

      The logic of this example applies to all systems of measurements that possess an addition.

      In general, when nothing is assumed in regards to commutativity of the operation, one regards the group operation as a multiplication. On the other hand, when the group operation is known to be commutative, one very often treats the group composition as an addition and uses a plus sign to designate the group composition.

      2. On page 480, in the third paragraph, you write “Since S3 restricts to an action on a subspace of R3, the action of S3 on R3 is reducible.” Does “restrict” here have the same meaning as “reduce”?

      Response:

      In general, the term “restricts” applies to functions or transformations, in which one specializes the application of the function to a sub-domain of the original function. For example, one might restrict the domain of the function y = x2 to apply only to numbers greater than or equal to zero.

      In your quoted excerpt, I am abusing the term “restrict” insofar as I also intend that the values of the S3 action are all within that same subspace. (One then says that the subspace is “invariant” under the S3 action.) Even with this (unintentional) abuse, the term “restrict” is not quite the same as “reduce,” because a “reduction” is a decomposition of the vector space into invariant subspaces. It is however, a fact that whenever one finds an invariant subspace, one can extend it to a decomposition of the original vector space into invariant subspaces. One proves, as a general proposition, that whenever a group action has an invariant subgroup, then the action is reducible.

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