# 3E: Exercises

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##### Exercise $$\PageIndex{1}$$
1. Consider the symmetric group $$S_4$$.

1. Find all the subgroup lattice of $$S_4$$, the symmetric group of order 24.

2. Find the centre of $$S_4$$.

3. Find the normalizer of each subgroup of $$S_4$$.

4. Find the cyclic subgroup of $$S_4$$ that has an order of 4.

5. Find a non-cyclic subgroup of $$S_4$$ that has order 4.

##### Exercise $$\PageIndex{2}$$
1. Consider the Dihedral group $$D_4$$.

1. Find all the subgroup lattice of $$D_4$$, the Dihedral group of order 8.

2. Find the centre of $$D_4$$.

3. Find the normalizer of each subgroup $$D_4$$.

4. Prove that every subgroup of $$D_4$$ of odd order is cyclic.

##### Exercise $$\PageIndex{3}$$
1. Prove that the following groups are non-abelian.

1. $$S_n$$, for $$n\ge 3$$.

2. $$A_n$$ for $$n \ge 4$$.

3. $$D_n$$, for $$n \ge 3$$.

##### Exercise $$\PageIndex{4}$$

Show that for $$n \ge 3$$, $$Z(S_n)=\{1\}$$.

##### Exercise $$\PageIndex{5}$$

a. Find a cyclic subgroup of $$A_8$$ that has order 4.

b. Find a non-cyclic subgroup of $$A_8$$ that has order 4.

##### Exercise $$\PageIndex{6}$$

Suppose that $$\sigma$$ and $$\tau$$ in $$S_5$$ satisfying  $$\tau \sigma = (1, 5, 2, 3)$$ and  $$\sigma \tau = (1, 2, 4, 5).$$ If  $$\sigma(1)=2,$$ find  $$\sigma$$ and  $$\tau.$$

##### Exercise $$\PageIndex{7}$$

Factor the permutation $$(1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3)$$ into disjoint cycles.

##### Exercise $$\PageIndex{8}$$

Let  $$\sigma = (k_1, k_2, \cdots, k_r)$$ be an $$r-$$cycle. If  $$\gamma \in S_n,$$ show that  $$\gamma \sigma \gamma^{-1}$$ is also a cycle; infact $$\gamma \sigma \gamma^{-1}= (\gamma(k_1), \gamma(k_2), \cdots, \gamma(k_r))$$

##### Exercise $$\PageIndex{9}$$

Show that every product of $$3-$$ cycles is an even permutation.

##### Exercise $$\PageIndex{10}$$

Show that every even permutation is a product of $$3-$$ cycles.

##### Exercise $$\PageIndex{11}$$
1. Let $$G$$ be a group. $$H=\{g^2: g\in G\}.$$ Prove or disprove:  $$H$$ is a subgroup.

(Hint: $$G=A_4.$$)

##### Exercise $$\PageIndex{12}$$

a. Find an element of maximum order in $$S_5$$.

b. Find an element of maximum order in $$S_7$$.

##### Exercise $$\PageIndex{13}$$

a. Let $$H$$ be a subgroup of $$A_4$$ containing $$K=\{e,(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\}$$. Shoe that if $$H$$ contains a $$3-$$ cycle then $$H=A_4.$$

b. Show that $$A_4$$ has no subgroup of order $$6.$$

This page titled 3E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.