3E: Exercises

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Nov 7, 2024
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- 3.3: Dihedral Groups (Group of Symmetries)
- Chapter 4: Cosets, special groups, and homorphism

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Exercise $\PageIndex{1}$

Compute the following permutations:

$(1435)(231)$
$143)(231)^2$
$(1345)^{-1}(234)(123)$
$(1234)^4$

Exercise $\PageIndex{2}$

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 $S_4$ with an order of 4.
5. Find a non-cyclic subgroup of $S_4$ with order 4.

Exercise $\PageIndex{3}$

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 $\PageIndex{4}$

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 $\PageIndex{5}$

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

Exercise $\PageIndex{6}$

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 $\PageIndex{7}$

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 $\PageIndex{8}$

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

2) Express the permutation $(1,2,3,4,5)(6,7)(1,3,5,7)(1,6,3)$ as products of transpositions, and decide if it is even or odd.

Exercise $\PageIndex{9}$

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 $\PageIndex{10}$

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

Exercise $\PageIndex{11}$

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

Exercise $\PageIndex{12}$

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

(Hint: $G=A_4.$ )

Exercise $\PageIndex{13}$

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

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

Exercise $\PageIndex{14}$

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

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

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Exercise 3E.1\PageIndex{1}

Exercise 3E.2\PageIndex{2}

Exercise 3E.3\PageIndex{3}

Exercise 3E.4\PageIndex{4}

Exercise 3E.5\PageIndex{5}

Exercise 3E.6\PageIndex{6}

Exercise 3E.7\PageIndex{7}

Exercise 3E.8\PageIndex{8}

Exercise 3E.9\PageIndex{9}

Exercise 3E.10\PageIndex{10}

Exercise 3E.11\PageIndex{11}

Exercise 3E.12\PageIndex{12}

Exercise 3E.13\PageIndex{13}

Exercise 3E.14\PageIndex{14}

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