Basic properties of the direct sum of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.17 Solution: (1) Note that $\prod 1 \in H$, where we may take $J = \emptyset$. Thus…
Basic properties of n-ary direct products of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…
Definition of the n-ary direct product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.15 Solution: (1) Elements in a direct product are uniquely represented, so well-definedness is not an issue.…
A quotient by a product is isomorphic to the product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.14 Solution: We begin with some lemmas. Lemma 1: Let $\varphi_1 : G_1 \rightarrow H_1$ and $\varphi_2…
Compute presentations for a given central product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.13 Solution: Note that $Z_4 \times D_8$ is generated by $(x,1)$, $(1,r)$, and $(1,s)$. Hence $Z_4 \ast_\varphi…
Basic properties of the central product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.12 Solution: (1) Let $\pi : A \times B \rightarrow (A \times B)/Z$ denote the canonical projection,…
Count the number of cyclic subgroups in an elementary abelian p-group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.11 Solution: By definition, every nonidentity element of $E_{p^n}$ has order $p$. Thus every nonidentity element generates…
Exhibit the distinct cyclic subgroups of an elementary abelian group of order $p^2$
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.10 Solution: We saw previously (by counting) that $E$ has precisely $p+1$ distinct subgroups of order $p$.…
Exhibit symmetric group as a subgroup of a general linear group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.9 Solution: We know that if $F$ is a finite field then $\mathsf{Aut}(F^n) \cong GL_n(F)$. This isomorphism…
Symmetric group acts on the n-fold direct product of a group by permuting the factors
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.8 Solution: First we show that this mapping $\psi : \pi \mapsto \varphi_\pi$ is a homomorphism. Let…