Chapter Questions
What is the smallest positive integer $n$ such that there are two nonisomorphic groups of order $n ?$ Name the two groups.
What is the smallest positive integer $n$ such that there are three nonisomorphic Abelian groups of order $n$ ? Name the three groups.
What is the smallest positive integer $n$ such that there are exactly four nonisomorphic Abelian groups of order $n ?$ Name the four groups.
Calculate the number of elements of order 2 in each of $Z_{16}, Z_{8} \oplus Z_{2}$, $Z_{4} \oplus Z_{4}$, and $Z_{4} \oplus Z_{2} \oplus Z_{2} .$ Do the same for the elements of order $4 .$
Prove that any Abelian group of order 45 has an element of order 15 . Does every Abelian group of order 45 have an element of order $9 ?$
Show that there are two Abelian groups of order 108 that have exactly one subgroup of order 3 .
Show that there are two Abelian groups of order 108 that have exactly four subgroups of order $3 .$
Show that there are two Abelian groups of order 108 that have exactly 13 subgroups of order $3 .$
Suppose that $G$ is an Abelian group of order 120 and that $G$ has exactly three elements of order 2. Determine the isomorphism class of $G$.
Find all Abelian groups (up to isomorphism) of order 360 .
Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic groups of orders $n_{1}, n_{2}, \ldots, n_{r}$, where $n_{i+1}$ divides $n_{i}$ for $i=1,2, \ldots, t-1$. (This exercise is referred to in this chapter.)
Suppose that the order of some finite Abelian group is divisible by 10. Prove that the group has a cyclic subgroup of order 10 .
Show, by example, that if the order of a finite Abelian group is divisible by 4 , the group need not have a cyclic subgroup of order $4 .$
On the basis of Exercises 12 and 13 , draw a general conclusion about the existence of cyclic subgroups of a finite Abelian group.
How many Abelian groups (up to isomorphism) are therea. of order $6 ?$b. of order $15 ?$c. of order $42 ?$d. of order $p q$, where $p$ and $q$ are distinct primes?e. of order $p q r$, where $p, q$, and $r$ are distinct primes?f. Generalize parts $\mathrm{d}$ and e.
How does the number (up to isomorphism) of Abelian groups of order $n$ compare with the number (up to isomorphism) of Abelian groups of order $m$ wherea. $n=3^{2}$ and $m=5^{2}$ ?b. $n=2^{4}$ and $m=5^{4}$ ?c. $n=p^{r}$ and $m=q^{r}$, where $p$ and $q$ are prime?d. $n=p^{r}$ and $m=p^{r} q$, where $p$ and $q$ are distinct primes?e. $n=p^{r}$ and $m=p^{r} q^{2}$, where $p$ and $q$ are distinct primes?
Up to isomorphism, how many additive Abelian groups of order 16 have the property that $x+x+x+x=0$ for all $x$ in the group?
Let $p_{1}, p_{2}, \ldots, p_{n}$ be distinct primes. Up to isomorphism, how many Abelian groups are there of order $p_{1}^{4} p_{2}^{4} \ldots p_{n}^{4}$ ?
The symmetry group of a nonsquare rectangle is an Abelian group of order 4 . Is it isomorphic to $Z_{4}$ or $Z_{2} \oplus Z_{2} ?$
Verify the corollary to the Fundamental Theorem of Finite Abelian Groups in the case that the group has order 1080 and the divisor is 180 .
The set $\{1,9,16,22,29,53,74,79,81\}$ is a group under multiplication modulo 91. Determine the isomorphism class of this group.
Suppose that $G$ is a finite Abelian group that has exactly one subgroup for each divisor of $|G| .$ Show that $G$ is cyclic.
Characterize those integers $n$ such that the only Abelian groups of order $n$ are cyclic.
Characterize those integers $n$ such that any Abelian group of order $n$ belongs to one of exactly four isomorphism classes.
Refer to Example 1 in this chapter and explain why it is unnecessary to compute the orders of the last five elements listed to determine the isomorphism class of $G$.
Let $G=\{1,7,17,23,49,55,65,71\}$ under multiplication modulo96. Express $G$ as an external and an internal direct product of cyclic groups.
Let $G=\{1,7,43,49,51,57,93,99,101,107,143,149,151,157$,$193,199\}$ under multiplication modulo 200. Express $G$ as an external and an internal direct product of cyclic groups.
The set $G=\{1,4,11,14,16,19,26,29,31,34,41,44\}$ is a group under multiplication modulo $45 .$ Write $G$ as an external and an internal direct product of cyclic groups of prime-power order.
Suppose that $G$ is an Abelian group of order $9 .$ What is the maximum number of elements (excluding the identity) of which one needs to compute the order to determine the isomorphism class of $G$ ? What if $G$ has order $18 ?$ What about $16 ?$
Suppose that $G$ is an Abelian group of order 16 , and in computing the orders of its elements, you come across an element of order 8 and two elements of order 2 . Explain why no further computations are needed to determine the isomorphism class of $G$.
Let $G$ be an Abelian group of order $16 .$ Suppose that there are elements $a$ and $b$ in $G$ such that $|a|=|b|=4$ and $a^{2} \neq b^{2}$. Determine the isomorphism class of $G$.
Prove that an Abelian group of order $2^{n}(n \geq 1)$ must have an odd number of elements of order $2 .$
Without using Lagrange's Theorem, show that an Abelian group of odd order cannot have an element of even order.
Let $G$ be the group of all $n \times n$ diagonal matrices with $\pm 1$ diagonal entries. What is the isomorphism class of $G$ ?
Prove the assertion made in the proof of Lemma 2 that $G=\langle a\rangle K$.
Suppose that $G$ is a finite Abelian group. Prove that $G$ has order $p^{n}$. where $p$ is prime, if and only if the order of every element of $G$ is a power of $p$.
Dirichlet's Theorem says that, for every pair of relatively prime integers $a$ and $b$, there are infinitely many primes of the form $a t+b$. Use Dirichlet's Theorem to prove that every finite Abelian group is isomorphic to a subgroup of a $U$ -group.
Determine the isomorphism class of $\operatorname{Aut}\left(Z_{2} \oplus Z_{3} \oplus Z_{5}\right)$.
Give an example to show that Lemma 2 is false if $G$ is non-Abelian.