struct.ipynb

Kernel:

%%html
<link href="http://mathbook.pugetsound.edu/beta/mathbook-content.css" rel="stylesheet" type="text/css" />
<link href="https://aimath.org/mathbook/mathbook-add-on.css" rel="stylesheet" type="text/css" />
<style>.subtitle {font-size:medium; display:block}</style>
<link href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css" />
<link href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext" rel="stylesheet" type="text/css" /><!-- Hide this cell. -->
<script>
var cell = $(".container .cell").eq(0), ia = cell.find(".input_area")
if (cell.find(".toggle-button").length == 0) {
ia.after(
    $('<button class="toggle-button">Toggle hidden code</button>').click(
        function (){ ia.toggle() }
        )
    )
ia.hide()
}
</script>

Important: to view this notebook properly you will need to execute the cell above, which assumes you have an Internet connection. It should already be selected, or place your cursor anywhere above to select. Then press the "Run" button in the menu bar above (the right-pointing arrowhead), or press Shift-Enter on your keyboard.

ParseError: KaTeX parse error: \newcommand{\lt} attempting to redefine \lt; use \renewcommand

The ultimate goal of group theory is to classify all groups up to isomorphism; that is, given a particular group, we should be able to match it up with a known group via an isomorphism. For example, we have already proved that any finite cyclic group of order $n$ is isomorphic to ${\mathbb Z}_n\text{;}$ hence, we “know” all finite cyclic groups. It is probably not reasonable to expect that we will ever know all groups; however, we can often classify certain types of groups or distinguish between groups in special cases.

In this chapter we will characterize all finite abelian groups. We shall also investigate groups with sequences of subgroups. If a group has a sequence of subgroups, say

$$\begin{equation*} G = H_n \supset H_{n - 1} \supset \cdots \supset H_1 \supset H_0 = \{ e \}, \end{equation*}$$

where each subgroup $H_i$ is normal in $H_{i+1}$ and each of the factor groups $H_{i+1}/H_i$ is abelian, then $G$ is a solvable group. In addition to allowing us to distinguish between certain classes of groups, solvable groups turn out to be central to the study of solutions to polynomial equations.

13.1Finite Abelian Groups

13.2Solvable Groups

13.3Exercises

13.4Programming Exercises

13.5References and Suggested Readings

13.6Sage

13.7Sage Exercises