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## 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,…

## 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 a group homomorphism from Z/(8) to Z/(4)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.10 Let $\varphi : \mathbb{Z}/(8) \rightarrow \mathbb{Z}/(4)$ be defined by $\overline{a} \mapsto \overline{a}$. (Note that $\overline{a}$ means…

## Characterize the automorphisms of cyclic group of order n

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.26 Let $Z_n = \langle \alpha \rangle$ be a cyclic group of order $n$ and for each…