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## Compute all Sylow subgroups of Dihedral group $D_{12}$ and $S_3\times S_3$

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: The Sylow 2-subgroups of $D_{12}$ have order 4. By Sylow’s Theorem, the number $n_2$ of…

## Use direct products to construct groups with some given properties

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.18 Solution: (1) Consider $\prod_{\mathbb{N}} \mathbb{Z}/(2)$; clearly this group is infinite, and moreover 2(\prod x_i) = \prod…

## 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.…