Homework 4

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Worksheet Homework 4

1.

Consider the group $(\Q^*,\cdot)\text{,}$ the non-zero rational numbers under multiplication. Let $H = \{2^k \st k \in \Z\}\text{.}$ Prove that $H$ is a subgroup of $G\text{.}$

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2.

Let $G$ be a group and let $H = \{x \in G \st x^2 = e\}\text{.}$

If $G = S_4\text{,}$ what is $H\text{?}$ Is $H$ a subgroup of $G\text{?}$
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Prove that if $G$ is abelian, then $H$ is a subgroup of $G\text{.}$
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Hint.

Warning: this is not the same as $G^2 = \{g^2 \st g \in G\}$ that we did in class.

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3.

Find a set of elements of $S_4$ that generates all of $S_4\text{.}$ Ensure that your set is as small as possible. Justify your answer (both that your set generates $S_4$ and that no smaller set does).

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Worksheet Homework 4

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Worksheet Homework 4