## Exercises 3.3 Exercises

###### 1.

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

This is true for every proper nontrivial subgroup.

###### 2.

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

False.

###### 3.

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 \(GL_2( {\mathbb R})\text{?}\)

###### 4.

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

\(4^{\phi(15)} \equiv 4^8 \equiv 1 \pmod{15}\text{.}\)

###### 5.

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

###### 6.

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

###### 7.

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

###### 8.

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.

\(\displaystyle g_1 H = g_2 H\)

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

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

\(\displaystyle g_2 \in g_1 H\)

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

###### 9.

What fails in the proof of Theorem 3.8 if \(\phi : {\mathcal L}_H \rightarrow {\mathcal R}_H\) is defined by \(\phi( gH ) = Hg\text{?}\)

###### 10.

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

###### 11.

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 \(\sigma = (12)(345)(78)(9)\) 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.

###### 12.

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.

###### 13.

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

Show that \(g(H \cap K) = gH \cap gK\text{.}\)

###### 14.

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 \(H = \{ (1),(123), (132) \}\) in \(A_4\text{.}\)

###### 15.

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

###### 16.

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

If \(\gcd(m,n) = 1\text{,}\) then \(\phi(mn) = \phi(m)\phi(n)\) ( Exercise 6.1.5.11).

###### 17.

Show that

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

###### 18.

For each of the following groups \(G\text{,}\) determine whether \(H\) is a normal subgroup of \(G\text{.}\) If \(H\) is a normal subgroup, write out a Cayley table for the factor group \(G/H\text{.}\)

\(G = S_4\) and \(H = A_4\)

\(G = A_5\) and \(H = \{ (1), (123), (132) \}\)

\(G = S_4\) and \(H = D_4\)

\(G = Q_8\) and \(H = \{ 1, -1, I, -I \}\)

\(G = {\mathbb Z}\) and \(H = 5 {\mathbb Z}\)

(a)

(c) \(D_4\) is not normal in \(S_4\text{.}\)

###### 19.

Find all the subgroups of \(D_4\text{.}\) Which subgroups are normal? What are all the factor groups of \(D_4\) up to isomorphism?

###### 20.

Find all the subgroups of the quaternion group, \(Q_8\text{.}\) Which subgroups are normal? What are all the factor groups of \(Q_8\) up to isomorphism?

###### 21.

Let \(T\) be the group of nonsingular upper triangular \(2 \times 2\) matrices with entries in \({\mathbb R}\text{;}\) that is, matrices of the form

where \(a\text{,}\) \(b\text{,}\) \(c \in {\mathbb R}\) and \(ac \neq 0\text{.}\) Let \(U\) consist of matrices of the form

where \(x \in {\mathbb R}\text{.}\)

Show that \(U\) is a subgroup of \(T\text{.}\)

Prove that \(U\) is abelian.

Prove that \(U\) is normal in \(T\text{.}\)

Show that \(T/U\) is abelian.

Is \(T\) normal in \(GL_2( {\mathbb R})\text{?}\)

###### 22.

Show that the intersection of two normal subgroups is a normal subgroup.

###### 23.

If \(G\) is abelian, prove that \(G/H\) must also be abelian.

###### 24.

Prove or disprove: If \(H\) is a normal subgroup of \(G\) such that \(H\) and \(G/H\) are abelian, then \(G\) is abelian.

###### 25.

If \(G\) is cyclic, prove that \(G/H\) must also be cyclic.

If \(a \in G\) is a generator for \(G\text{,}\) then \(aH\) is a generator for \(G/H\text{.}\)

###### 26.

Prove or disprove: If \(H\) and \(G/H\) are cyclic, then \(G\) is cyclic.

###### 27.

Let \(H\) be a subgroup of index \(2\) of a group \(G\text{.}\) Prove that \(H\) must be a normal subgroup of \(G\text{.}\) Conclude that \(S_n\) is not simple for \(n \geq 3\text{.}\)

###### 28.

If a group \(G\) has exactly one subgroup \(H\) of order \(k\text{,}\) prove that \(H\) is normal in \(G\text{.}\)

For any \(g \in G\text{,}\) show that the map \(i_g : G \to G\) defined by \(i_g : x \mapsto gxg^{-1}\) is an isomorphism of \(G\) with itself. Then consider \(i_g(H)\text{.}\)

###### 29.

Define the centralizer of an element \(g\) in a group \(G\) to be the set

Show that \(C(g)\) is a subgroup of \(G\text{.}\) If \(g\) generates a normal subgroup of \(G\text{,}\) prove that \(C(g)\) is normal in \(G\text{.}\)

Suppose that \(\langle g \rangle\) is normal in \(G\) and let \(y\) be an arbitrary element of \(G\text{.}\) If \(x \in C(g)\text{,}\) we must show that \(y x y^{-1}\) is also in \(C(g)\text{.}\) Show that \((y x y^{-1}) g = g (y x y^{-1})\text{.}\)

###### 30.

Recall that the center of a group \(G\) is the set

Calculate the center of \(S_3\text{.}\)

Calculate the center of \(GL_2 ( {\mathbb R} )\text{.}\)

Show that the center of any group \(G\) is a normal subgroup of \(G\text{.}\)

If \(G / Z(G)\) is cyclic, show that \(G\) is abelian.

###### 31.

Let \(G\) be a group and let \(G' = \langle aba^{- 1} b^{-1} \rangle\text{;}\) that is, \(G'\) is the subgroup of all finite products of elements in \(G\) of the form \(aba^{-1}b^{-1}\text{.}\) The subgroup \(G'\) is called the commutator subgroup of \(G\text{.}\)

Show that \(G'\) is a normal subgroup of \(G\text{.}\)

Let \(N\) be a normal subgroup of \(G\text{.}\) Prove that \(G/N\) is abelian if and only if \(N\) contains the commutator subgroup of \(G\text{.}\)

(a) Let \(g \in G\) and \(h \in G'\text{.}\) If \(h = aba^{-1}b^{-1}\text{,}\) then

We also need to show that if \(h = h_1 \cdots h_n\) with \(h_i = a_i b_i a_i^{-1} b_i^{-1}\text{,}\) then \(ghg^{-1}\) is a product of elements of the same type. However, \(ghg^{-1} = g h_1 \cdots h_n g^{-1} = (gh_1g^{-1})(gh_2g^{-1}) \cdots (gh_ng^{-1})\text{.}\)