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…
Finite direct products are isomorphic up to permutation of the factors
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.7 Solution: We need to show that $\varphi_\pi$ is a bijective homomorphism. Homomorphism: Let $g = (g_i)$…
In the direct product of the quaternion group and an elementary abelian 2-group, all subgroups are normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.6 Solution: Let $H \leq Q_8 \times E_{2^k}$ be a subgroup, and let $\alpha = (a,x) \in…
The direct product of groups with no nonnormal subgroups may have nonnormal subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.5 Solution: Write $Z_4 = \langle x \rangle$ and consider $\langle (i,x) \rangle = \{ (i,x), (-1,x^2),…
The Sylow numbers of a direct product are products of the Sylow numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.4 Solution: Let $A$ and $B$ be finite groups and $p$ a prime, and write $|A| =…
Elements of generalized coordinate subgroups commute pairwise
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.3 Solution: Let $x = (x_j)$ and $y = (y_j)$. If $j \in I$, then $$(xy)_j =…