2.8: Exercises, Part II

1. True/False. For each of the following, write T if the statement is true; otherwise, write F. You do NOT need to provide explanations or show work for this problem.

1. For every positive integer \(n\text{,}\) there exists a group of order \(n\text{.}\)
2. For every integer \(n\geq 2\text{,}\) \(\mathbb{Z}_n\) is abelian.
3. Every abelian group is finite.
4. For every integer \(m\) and integer \(n\geq 2\text{,}\) there exist infinitely many integers \(a\) such that \(a\) is congruent to \(m\) modulo \(n\text{.}\)
5. A binary operation \(*\) on a set \(S\) is commutative if and only if there exist \(a,b\in S\) such that \(a*b=b*a\text{.}\)
6. If \(\langle S, *\rangle\) is a binary structure, then the elements of \(S\) must be numbers.
7. If \(e\) is an identity element of a binary structure (not necessarily a group) \(\langle S,*\rangle\text{,}\) then \(e\) is an idempotent in \(S\) (that is, \(e*e=e\)).
8. If \(s\) is an idempotent in a binary structure (not necessarily a group) \(\langle S,*\rangle\text{,}\) then \(s\) must be an identity element of \(S\text{.}\)

2. Let \(G\) be the set of all functions from \(\mathbb{Z}\) to \(\mathbb{R}\text{.}\) Prove that pointwise multiplication on \(G\) (that is, the operation defined by \((fg)(x)=f(x)g(x)\) for all \(f,g\in G\) and \(x\in \mathbb{Z}\)) is commutative. (Note. To prove that two functions, \(h\) and \(j\text{,}\) sharing the same domain \(D\) are equal, you need to show that \(h(x)=j(x)\) for every \(x\in D\text{.}\))

3. Decide which of the following binary structures are groups. For each, if the binary structure isn't a group, prove that. (Remember, you should not state that inverses do or do not exist for elements until you have made sure that the structure contains an identity element!) If the binary structure is a group, prove that.

a. \(\mathbb{Q}\) under multiplication

b. \(\mathbb{M}_2(\mathbb{R})\) under addition

c. \(\mathbb{M}_2(\mathbb{R})\) under multiplication

d. \(\mathbb{R}^+\) under \(*\text{,}\) defined by \(a*b=\sqrt{ab}\) for all \(a,b\in \mathbb{R}^+\)

4. Give an example of an abelian group containing 711 elements.

5. Let \(n\in \mathbb{Z}\text{.}\) Prove that \(n\mathbb{Z}\) is a group under the usual addition of integers. Note: You may use the fact that \(\langle n\mathbb{Z},+\rangle\) is a binary structure if you provide a reference for this fact.

6. Let \(n\in \mathbb{Z}^+\text{.}\) Prove that \(SL(n,\mathbb{R})\) is a group under matrix multiplication. Note: You may use the fact that \(\langle SL(n\mathbb{R}),\cdot\rangle\) is a binary structure if you provide a reference for this fact.
7.  

1. List three distinct integers that are congruent to \(6\) modulo \(5\text{.}\)

2. List the elements of \(\mathbb{Z}_5\text{.}\)

3. Compute:

   1. \(4+5\) in \(\mathbb{Z}\text{;}\)

   2. \(4+5\) in \(\mathbb{Q}\text{;}\)

   3. \(4+_65\) in \(\mathbb{Z}_6\text{;}\)

   4. the inverse of \(4\) in \(\mathbb{Z}\text{;}\)

   5. the inverse of \(4\) in \(\mathbb{Z}_6\text{.}\)

4. Why does it not make sense for me to ask you to compute \(4+_3 2\) in \(\mathbb{Z}_3\text{?}\) Please answer this using a complete, grammatically correct sentence.

8. Let \(G\) be a group with identity element \(e\text{.}\) Prove that if every element of \(G\) is its own inverse, then \(G\) is abelian.

9. Let \(G\) be a group. The subset

\begin{equation*} Z(G):=\{z \in G\,:\, zg=gz \mbox{ for all } g\in G\} \end{equation*}

of \(G\) is called the center of \(G\text{.}\) In other words, \(Z(G)\) is the set of all elements of \(G\) that commute with every element of \(G\text{.}\) Prove that \(Z(G)\) is closed in \(G\text{.}\)