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## Exhibit two Sylow 2-subgroups of Symmetric group $S_5$

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.8 Solution: $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, so that the Sylow 2-subgroups of…

## Exhibit Dihedral group as a subgroup of Symmetric group via regular representation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.3 Solution: To save effort we will perform this computation in $S_{D_8}$ and then use the labeling…

## Exhibit Sym(3) as a subgroup of Sym(6) via the left regular representation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.2 Solution: We use the notation $\sigma(k) = \sigma \cdot k$. (1) $1 \mapsto 1$ (2) We…

## Exhibit the Klein 4-group as a subgroup of Sym(4) using the left regular representation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.1 Solution: The multiplication table for $G$ is as follows.    1 a b c 1 1…

## 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…