## Exercises 6.5 Exercises

###### 1.

Suppose that \(G\) is a finite group with an element \(g\) of order \(5\) and an element \(h\) of order \(7\text{.}\) Why must \(|G| \geq 35\text{?}\)

###### 2.

Suppose that \(G\) is a finite group with \(60\) elements. What are the orders of possible subgroups of \(G\text{?}\)

###### 3.

Prove or disprove: Every subgroup of the integers has finite index.

###### 4.

Prove or disprove: Every subgroup of the integers has finite order.

###### 5.

List the left and right cosets of the subgroups in each of the following.

\(\langle 8 \rangle\) in \({\mathbb Z}_{24}\)

\(\langle 3 \rangle\) in \(U(8)\)

\(3 {\mathbb Z}\) in \({\mathbb Z}\)

\(A_4\) in \(S_4\)

\(A_n\) in \(S_n\)

\(D_4\) in \(S_4\)

\({\mathbb T}\) in \({\mathbb C}^\ast\)

\(H = \{ (1), (1 \, 2 \, 3), (1 \, 3 \, 2) \}\) in \(S_4\)

###### 6.

Describe the left cosets of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\text{.}\) What is the index of \(SL_2( {\mathbb R} )\) in \(GL_2( {\mathbb R})\text{?}\)

###### 7.

Verify Euler's Theorem for \(n = 15\) and \(a = 4\text{.}\)

###### 8.

Use Fermat's Little Theorem to show that if \(p = 4n + 3\) is prime, there is no solution to the equation \(x^2 \equiv -1 \pmod{p}\text{.}\)

###### 9.

Show that the integers have infinite index in the additive group of rational numbers.

###### 10.

Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.

###### 11.

Let \(H\) be a subgroup of a group \(G\) and suppose that \(g_1, g_2 \in G\text{.}\) Prove that the following conditions are equivalent.

\(\displaystyle g_1 H = g_2 H\)

\(\displaystyle H g_1^{-1} = H g_2^{-1}\)

\(\displaystyle g_1 H \subset g_2 H\)

\(\displaystyle g_2 \in g_1 H\)

\(\displaystyle g_1^{-1} g_2 \in H\)

###### 12.

If \(ghg^{-1} \in H\) for all \(g \in G\) and \(h \in H\text{,}\) show that right cosets are identical to left cosets. That is, show that \(gH = Hg\) for all \(g \in G\text{.}\)

###### 13.

What fails in the proof of TheoremĀ 6.8 if \(\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H\) is defined by \(\phi( gH ) = Hg\text{?}\)

###### 14.

Suppose that \(g^n = e\text{.}\) Show that the order of \(g\) divides \(n\text{.}\)

###### 15.

The cycle structure of a permutation \(\sigma\) is defined as the unordered list of the sizes of the cycles in the cycle decomposition \(\sigma\text{.}\) For example, the permutation \(\sigma = (1 \, 2)(3 \, 4 \, 5)(7 \, 8)(9)\) has cycle structure \((2,3,2,1)\) which can also be written as \((1, 2, 2, 3)\text{.}\)

Show that any two permutations \(\alpha, \beta \in S_n\) have the same cycle structure if and only if there exists a permutation \(\gamma\) such that \(\beta = \gamma \alpha \gamma^{-1}\text{.}\) If \(\beta = \gamma \alpha \gamma^{-1}\) for some \(\gamma \in S_n\text{,}\) then \(\alpha\) and \(\beta\) are conjugate.

###### 16.

If \(|G| = 2n\text{,}\) prove that the number of elements of order \(2\) is odd. Use this result to show that \(G\) must contain a subgroup of order 2.

###### 17.

Suppose that \([G : H] = 2\text{.}\) If \(a\) and \(b\) are not in \(H\text{,}\) show that \(ab \in H\text{.}\)

###### 18.

If \([G : H] = 2\text{,}\) prove that \(gH = Hg\text{.}\)

###### 19.

Let \(H\) and \(K\) be subgroups of a group \(G\text{.}\) Prove that \(gH \cap gK\) is a coset of \(H \cap K\) in \(G\text{.}\)

###### 20.

Let \(H\) and \(K\) be subgroups of a group \(G\text{.}\) Define a relation \(\sim\) on \(G\) by \(a \sim b\) if there exists an \(h \in H\) and a \(k \in K\) such that \(hak = b\text{.}\) Show that this relation is an equivalence relation. The corresponding equivalence classes are called double cosets. Compute the double cosets of \(H = \{ (1),(1 \, 2 \, 3), (1 \, 3 \, 2) \}\) in \(A_4\text{.}\)

###### 21.

Let \(G\) be a cyclic group of order \(n\text{.}\) Show that there are exactly \(\phi(n)\) generators for \(G\text{.}\)

###### 22.

Let \(n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}\text{,}\) where \(p_1, p_2, \ldots, p_k\) are distinct primes. Prove that

###### 23.

Show that

for all positive integers \(n\text{.}\)