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## 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 $p$.…

## Every abelian simple group has prime order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: Let $G$ be an abelian simple group. Suppose $G$ is infinite. If $x \in G$…

## The center of a direct product is the direct product of the centers

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.1 Show that the center of a direct product is the direct product of the centers: Z(G_1…

## An abelian group has the same cardinality as any sets on which it acts transitively

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.3 Suppose $G \leq S_A$ is an abelian and transitive subgroup. Show that $\sigma(a) \neq a$ for…

## The intersection by an abelian normal subgroup is normal in the product

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.20 Let $G$ be a group and $A,B \leq G$ be subgroups such that $A$ is abelian…

## A finite group of width two has a trivial center

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.4 Let $G$ be a group. Prove that if $|G| = pq$ for some primes $p$ and…

## The set of all endomorphisms of an abelian group is a ring under pointwise addition and composition

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.29 Let $A$ be any abelian group. Let $R = \mathsf{Hom}(A,A)$ be the set of all group…

## Q/Z is divisible

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15 Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible.…

## Every quotient of an abelian group is abelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.3 Let $A$ be an abelian group and let $B \leq A$. Prove that $A/B$ is abelian.…

## Additive subgroups of the rationals which are closed under inversion are trivial

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.13 Let $H$ be a subgroup of the additive group of rational numbers with the property that…