The Binomial Theorem holds in any commutative ring
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.25 Solution: We begin with some lemmas. Recall that ${n \choose k} = \frac{n!}{k!(n-k)!}$, where $n$ is…
Powers in a quotient group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.4 Let $G$ be a group and $N$ a normal subgroup of $G$. Show that for all…
Group homomorphisms preserve exponents
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.1 Let $G$ and $H$ be groups and $\varphi : G \rightarrow H$ a group homomorphism. (1)…
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