10.3: Reading Questions

Abstract Algebra: Theory and Applications · LibreTexts

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    From LibreTexts, under GNU Free Documentation, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    1

    Let \(G\) be the group of symmetries of an equilateral triangle, expressed as permutations of the vertices numbered \(1,2,3\text{.}\) Let \(H\) be the subgroup \(H=\langle (1\,2) \rangle\text{.}\) Build the left and right cosets of \(H\) in \(G\text{.}\)

    2

    Based on your answer to the previous question, is \(H\) normal in \(G\text{?}\) Explain why or why not.

    3

    The subgroup \(8\mathbb Z\) is normal in \(\mathbb Z\text{.}\) In the factor group \(\mathbb Z/8\mathbb Z\) perform the computation \((3+8\mathbb Z)+(7+8\mathbb Z)\text{.}\)

    4

    List two statements about a group \(G\) and a subgroup \(H\) that are equivalent to “\(H\) is normal in \(G\text{.}\)”

    5

    In your own words, what is a factor group?

    This page titled 10.3: Reading Questions was last modified on Tue, 01 Sep 2026 06:45:28 GMT and is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.