Chapter 6

For a finite abelian group, one can completely specify the group by writing down the group operation table. For instance, Example 2.7 presented an addition table for \(\mathbb{Z}_{6}\) (a) Write down group operation tables for the following finite abelian groups: \(\mathbb{Z}_{5}, \mathbb{Z}_{5}^{*},\) and \(\mathbb{Z}_{3} \times \mathbb{Z}_{4}^{*}\) (b) Show that the group operation table for every finite abelian group is a Latin square; that is, each element of the group appears exactly once in each row and column. (c) Below is an addition table for an abelian group that consists of the elements \(\\{a, b, c, d\\} ;\) however, some entries are missing. Fill in the missing entries.

Let \(G:=\\{x \in \mathbb{R}: x>1\\},\) and define \(a \star b:=a b-a-b+2\) for all \(a, b \in \mathbb{R}\). Show that: (a) \(G\) is closed under \(\star\); (b) the set \(G\) under the operation \(\star\) forms an abelian group.

Let \(G\) be an abelian group, and let \(g\) be an arbitrary, fixed element of \(G\). Assume that the group operation of \(G\) is written additively. We define a new binary operation \(\odot\) on \(G,\) as follows: for \(a, b \in G,\) let \(a \odot b:=a+b+g .\) Show that the set \(G\) under \(\odot\) forms an abelian group.

Let \(G\) be an abelian group, and let \(g\) be an arbitrary, fixed element of \(G\). Assume that the group operation of \(G\) is written additively. We define a new binary operation \(\odot\) on \(G,\) as follows: for \(a, b \in G,\) let \(a \odot b:=a+b+g .\) Show that the set \(G\) under \(\odot\) forms an abelian group.

Let \(\star\) be a binary operation on a non-empty, finite set \(G\). Assume that \(\star\) is associative, commutative, and satisfies the cancellation law: \(a \star b=a \star c\) implies \(b=c\). Show that \(G\) under \(\star\) forms an abelian group.

Let \(G\) be an abelian group. (a) Suppose that \(H\) is a non-empty subset of \(G\). Show that \(H\) is a subgroup of \(G\) if and only if \(a-b \in H\) for all \(a, b \in H\). (b) Suppose that \(H\) is a non-empty, finite subset of \(G\) such that \(a+b \in H\) for all \(a, b \in H .\) Show that \(H\) is a subgroup of \(G\).

Let \(G\) be an abelian group. (a) Show that if \(H\) is a subgroup of \(G, h \in H,\) and \(g \in G \backslash H,\) then \(h+g \in G \backslash H\) (b) Suppose that \(H\) is a non-empty subset of \(G\) such that for all \(h, g \in G:\) (i) \(h \in H\) implies \(-h \in H,\) and (ii) \(h \in H\) and \(g \in G \backslash H\) implies \(h+g \in G \backslash H\). Show that \(H\) is a subgroup of \(G\)

Show that if \(H\) is a subgroup of an abelian group \(G,\) then a set \(K \subseteq H\) is a subgroup of \(G\) if and only if \(K\) is a subgroup of \(H\).

Let \(G\) be an abelian group with subgroups \(H_{1}\) and \(H_{2}\). Show that every subgroup \(H\) of \(G\) that contains \(H_{1} \cup H_{2}\) must contain all of \(H_{1}+H_{2}\), and that \(H_{1} \subseteq H_{2}\) if and only if \(H_{1}+H_{2}=H_{2}\).

Let \(H_{1}\) be a subgroup of an abelian group \(G_{1}\) and \(H_{2}\) a subgroup of an abelian group \(G_{2}\). Show that \(H_{1} \times H_{2}\) is a subgroup of \(G_{1} \times G_{2}\).

Let \(H_{1}\) be a subgroup of an abelian group \(G_{1}\) and \(H_{2}\) a subgroup of an abelian group \(G_{2}\). Show that \(H_{1} \times H_{2}\) is a subgroup of \(G_{1} \times G_{2}\).

Let \(G_{1}\) and \(G_{2}\) be abelian groups, and let \(H\) be a subgroup of \(G_{1} \times G_{2} .\) Define $$ H_{1}:=\left\\{a_{1} \in G_{1}:\left(a_{1}, a_{2}\right) \in H \text { for some } a_{2} \in G_{2}\right\\} $$ Show that \(H_{1}\) is a subgroup of \(G_{1}\).

}(I, G)\( be the set of f… # Let \)I\( be a set and \)G\( be an abelian group, and consider the group \)\operatorname{Map}(I, G)\( of functions \)f: I \rightarrow G .\( Let Map \)^{\\#}(I, G)\( be the set of functions \)f \in \operatorname{Map}(I, G)\( such that \)f(i) \neq 0_{G}\( for at most finitely many \)i \in I\(. Show that \)\operatorname{Map}^{\\#}(I, G)\( is a subgroup of \)\operatorname{Map}(I, G)$

Write down the cosets of \(\left(\mathbb{Z}_{35}^{*}\right)^{2}\) in \(\mathbb{Z}_{35}^{*}\), along with the multiplication table for the quotient group \(\mathbb{Z}_{35}^{*} /\left(\mathbb{Z}_{35}^{*}\right)^{2}\).

Let \(n\) be an odd, positive integer whose factorization into primes is \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}} .\) Show that \(\left[\mathbb{Z}_{n}^{*}:\left(\mathbb{Z}_{n}^{*}\right)^{2}\right]=2^{r}\).

Let \(n\) be a positive integer, and let \(m\) be any integer. Show that \(\left[\mathbb{Z}_{n}: m \mathbb{Z}_{n}\right]=n / \operatorname{gcd}(m, n)\)

Let \(G\) be an abelian group and \(H\) a subgroup with \([G: H]=2 .\) Show that if \(a, b \in G \backslash H,\) then \(a+b \in H\).

Let \(H\) be a subgroup of an abelian group \(G,\) and let \(a, b \in G\) with \(a \equiv b(\bmod H) .\) Show that \(k a \equiv k b(\bmod H)\) for all \(k \in \mathbb{Z}\).

Let \(G\) be an abelian group, and let \(\sim\) be an equivalence relation on \(G\). Further, suppose that for all \(a, a^{\prime}, b \in G,\) if \(a \sim a^{\prime},\) then \(a+b \sim a^{\prime}+b .\) Let \(H:=\left\\{a \in G: a \sim 0_{G}\right\\} .\) Show that \(H\) is a subgroup of \(G,\) and that for all \(a, b \in G\) we have \(a \sim b\) if and only if \(a \equiv b(\bmod H)\).

Let \(H\) be a subgroup of an abelian group \(G,\) and let \(a, b \in G\) Show that \([a+b]_{H}=\left\\{x+y: x \in[a]_{H}, y \in[b]_{H}\right\\}\)

Verify that the "is isomorphic to" relation on abelian groups is an equivalence relation; that is, for all abelian groups \(G_{1}, G_{2}, G_{3},\) we have: (a) \(G_{1} \cong G_{1}\) (b) \(G_{1} \cong G_{2}\) implies \(G_{2} \cong G_{1}\); (c) \(G_{1} \cong G_{2}\) and \(G_{2} \cong G_{3}\) implies \(G_{1} \cong G_{3}\).

Let \(\rho_{i}: G_{i} \rightarrow G_{i}^{\prime},\) for \(i=1, \ldots, k,\) be group homomorphisms. Show that the map $$ \begin{aligned} \rho: \quad G_{1} \times \cdots \times G_{k} & \rightarrow G_{1}^{\prime} \times \cdots \times G_{k}^{\prime} \\ \left(a_{1}, \ldots, a_{k}\right) & \mapsto\left(\rho_{1}\left(a_{1}\right), \ldots, \rho_{k}\left(a_{k}\right)\right) \end{aligned} $$ is a group homomorphism. Also show that if each \(\rho_{i}\) is an isomorphism, then so is \(\rho .\)

Let \(\rho: G \rightarrow G^{\prime}\) be a group homomorphism. Let \(H, K\) be subgroups of \(G\) and let \(m\) be a positive integer. Show that \(\rho(H+K)=\rho(H)+\rho(K)\) and \(\rho(m H)=m \rho(H) .\)

Let \(\rho: G \rightarrow G^{\prime}\) be a group homomorphism. Let \(H\) be a subgroup of \(G,\) and let \(\tau: H \rightarrow G^{\prime}\) be the restriction of \(\rho\) to \(H\). Show that \(\tau\) is a group homomorphism and that \(\operatorname{Ker} \tau=\operatorname{Ker} \rho \cap H\).

Suppose \(G_{1}, \ldots, G_{k}\) are abelian groups. Show that for each \(i=1, \ldots, k,\) the projection map \(\pi_{i}: G_{1} \times \cdots \times G_{k} \rightarrow G_{i}\) that sends \(\left(a_{1}, \ldots, a_{k}\right)\) to \(a_{i}\) is a surjective group homomorphism.

Show that if \(G=G_{1} \times G_{2}\) for abelian groups \(G_{1}\) and \(G_{2},\) and \(H_{1}\) is a subgroup of \(G_{1}\) and \(H_{2}\) is a subgroup of \(G_{2},\) then we have a group isomorphism \(G /\left(H_{1} \times H_{2}\right) \cong G_{1} / H_{1} \times G_{2} / H_{2}\).

Let \(G\) be an abelian group with subgroups \(H\) and \(K\). (a) Show that we have a group isomorphism \((H+K) / K \cong H /(H \cap K)\). (b) Show that if \(H\) and \(K\) are finite, then \(|H+K|=|H \| K| /|H \cap K| .\)

Let \(\rho: G \rightarrow G^{\prime}\) be a group homomorphism with kernel \(K .\) Let \(H\) be a subgroup of \(G\). Show that we have a group isomorphism \(G /(H+K) \cong\) \(\rho(G) / \rho(H)\)

Let \(\rho: G \rightarrow G^{\prime}\) be a surjective group homomorphism. Let \(S\) be the set of all subgroups of \(G\) that contain Ker \(\rho,\) and let \(S^{\prime}\) be the set of all subgroups of \(G^{\prime} .\) Show that the sets \(\mathcal{S}\) and \(\mathcal{S}^{\prime}\) are in one-to-one correspondence, via the map that sends \(H \in \mathcal{S}\) to \(\rho(H) \in \mathcal{S}^{\prime} .\) Also show that this correspondence preserves inclusions; that is, for all \(H_{1}, H_{2} \in \mathcal{S},\) we have \(H_{1} \subseteq H_{2} \Longleftrightarrow \rho\left(H_{1}\right) \subseteq \rho\left(H_{2}\right) .\)

Suppose that \(G, G_{1},\) and \(G_{2}\) are abelian groups, and that \(\rho\) : \(G_{1} \times G_{2} \rightarrow G\) is a group isomorphism. Let \(H_{1}:=\rho\left(G_{1} \times\left\\{0_{G_{2}}\right\\}\right)\) and \(H_{2}:=\) \(\rho\left(\left\\{0_{G_{1}}\right\\} \times G_{2}\right) .\) Show that \(G\) is the internal direct product of \(H_{1}\) and \(H_{2}\).

Let \(\mathbb{Z}^{+}\) denote the set of positive integers, and let \(\mathbb{Q}^{*}\) be the multiplicative group of non-zero rational numbers. Consider the abelian groups Map \(^{\\#}\left(\mathbb{Z}^{+}, \mathbb{Z}\right)\) and Map \(^{\\#}\left(\mathbb{Z}^{+}, \mathbb{Z}_{2}\right),\) as defined in Exercise \(6.14 .\) Show that we have group isomorphisms (a) \(\mathbb{Q}^{*} \cong \mathbb{Z}_{2} \times \operatorname{Map}^{\\#}\left(\mathbb{Z}^{+}, \mathbb{Z}\right),\) and (b) \(\mathbb{Q}^{*} /\left(\mathbb{Q}^{*}\right)^{2} \cong \operatorname{Map}^{\\#}\left(\mathbb{Z}^{+}, \mathbb{Z}_{2}\right)\).

Let \(n\) be an odd, positive integer whose factorization into primes is \(n=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}} .\) Show that: (a) we have a group isomorphism \(\mathbb{Z}_{n}^{*} /\left(\mathbb{Z}_{n}^{*}\right)^{2} \cong \mathbb{Z}_{2}^{\times r}\); (b) if \(p_{i} \equiv 3(\bmod 4)\) for each \(i=1, \ldots, r,\) then the squaring map on \(\left(\mathbb{Z}_{n}^{*}\right)^{2}\) is a group automorphism.

Which of the following pairs of groups are isomorphic? Why or (d) \(\mathbb{Z}_{2} \times \mathbb{Z}\) and \(\mathbb{Z},(\mathrm{e})\) why not? (a) \(\mathbb{Z}_{2} \times \mathbb{Z}_{2}\) and \(\mathbb{Z}_{4},\) (b) \(\mathbb{Z}_{12}^{*}\) and \(\mathbb{Z}_{8}^{*},\) (c) \(\mathbb{Z}_{5}^{*}\) and \(\mathbb{Z}_{4}\), \(\mathbb{Q}\) and \(\mathbb{Z},\) (f) \(\mathbb{Z} \times \mathbb{Z}\) and \(\mathbb{Z}\)

Show that \(\mathbb{Q}^{*}\) is not finitely generated.

Let \(G\) be an abelian group, \(a \in G,\) and \(m \in \mathbb{Z},\) such that \(m>0\) and \(m a=0_{G} .\) Let \(m=p_{1}^{e_{1}} \cdots p_{r}^{e_{r}}\) be the prime factorization of \(m .\) For \(i=1, \ldots, r\), let \(f_{i}\) be the largest non-negative integer such that \(f_{i} \leq e_{i}\) and \(m / p_{i}^{f_{i}} \cdot a=0_{G}\) Show that the order of \(a\) is equal to \(p_{1}^{e_{1}-f_{1}} \cdots p_{r}^{e_{r}-f_{r}}\).

Let \(G\) be an abelian group of order \(n,\) and let \(m\) be an integer. Show that \(m G=G\) if and only if \(\operatorname{gcd}(m, n)=1\)

Let \(H\) be a subgroup of an abelian group \(G .\) Show that: (a) if \(H\) and \(G / H\) are both finitely generated, then so is \(G\); (b) if \(G\) is finite, \(\operatorname{gcd}(|H|,|G / H|)=1,\) and \(H\) and \(G / H\) are both cyclic, then \(G\) is cyclic.

Let \(G\) be an abelian group of exponent \(m_{1} m_{2},\) where \(m_{1}\) and \(m_{2}\) are relatively prime. Show that \(G\) is the internal direct product of \(m_{1} G\) and \(m_{2} G\).

As additive groups, \(\mathbb{Z}\) is clearly a subgroup of \(\mathbb{Q} .\) Consider the quotient group \(G:=\mathbb{Q} / \mathbb{Z},\) and show that: (a) all elements of \(G\) have finite order; (b) \(G\) has exponent 0 ; (c) for all positive integers \(m,\) we have \(m G=G\) and \(G\\{m\\} \cong \mathbb{Z}_{m} ;\) (d) all finite subgroups of \(G\) are cyclic.

Suppose that \(G\) is an abelian group that satisfies the following properties: (i) for all \(m \in \mathbb{Z}, G\\{m\\}\) is either equal to \(G\) or is of finite order; (ii) for some \(m \in \mathbb{Z},\left\\{0_{G}\right\\} \subsetneq G\\{m\\} \subsetneq G\). Show that \(G\\{m\\}\) is finite for all non-zero \(m \in \mathbb{Z}\).

In our proof of Euler's criterion (Theorem 2.21 ), we really only used the fact that \(\mathbb{Z}_{p}^{*}\) has a unique element of multiplicative order \(2 .\) This exercise develops a proof of a generalization of Euler's criterion, based on the fundamental theorem of finite abelian groups. Suppose \(G\) is an abelian group of even order \(n\) that contains a unique element of order 2 . (a) Show that \(G \cong \mathbb{Z}_{2} \times \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{k}},\) where \(e>0\) and the \(m_{i}\) 's are odd integers. (b) Using part (a), show that \(2 G=G\\{n / 2\\}\).

Let \(G\) be a non-trivial, finite abelian group. Let \(s\) be the smallest positive integer such that \(G=\left\langle a_{1}, \ldots, a_{s}\right\rangle\) for some \(a_{1}, \ldots, a_{s} \in G\). Show that \(s\) is equal to the value of \(t\) in Theorem \(6.45 .\) In particular, \(G\) is cyclic if and only if \(t=1\)

Suppose \(G \cong \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{t}} .\) Let \(p\) be a prime, and let \(s\) be the number of \(m_{i}\) 's divisible by \(p\). Show that \(G\\{p\\} \cong \mathbb{Z}_{p}^{\times s}\).

Suppose \(G \cong \mathbb{Z}_{m_{1}} \times \cdots \times \mathbb{Z}_{m_{t}}\) with \(m_{i} \mid m_{i+1}\) for \(i=1, \ldots, t-1\), and that \(H\) is a subgroup of \(G .\) Show that \(H \cong \mathbb{Z}_{n_{1}} \times \cdots \times \mathbb{Z}_{n_{t}},\) where \(n_{i} \mid n_{i+1}\) for \(i=1, \ldots, t-1\) and \(n_{i} \mid m_{i}\) for \(i=1, \ldots, t\)

Suppose that \(G\) is an abelian group such that for all \(m>0\), we have \(m G=G\) and \(|G\\{m\\}|=m^{2}\) (note that \(G\) is not finite). Show that \(G\\{m\\} \cong \mathbb{Z}_{m} \times \mathbb{Z}_{m}\) for all \(m>0 .\)