Abstract Algebra: Theory and Applications, Remixed!

1.

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

2.

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

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 \(a = 4\text{.}\)

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.

1. \(\displaystyle g_1 H = g_2 H\)

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

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

4. \(\displaystyle g_2 \in g_1 H\)

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

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 \(G\text{.}\)

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

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 \(G/H\text{.}\)

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

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

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

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

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

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

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

2. Prove that \(U\) is abelian.

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

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

5. 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.

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 \(G\text{.}\)

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 \(G\text{.}\)

30.

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

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

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

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

4. 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 \(G\text{.}\)

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

2. 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 \(G\text{.}\)