In a p-group, every proper subgroup of minimal index is normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.9 Solution: Let $G$ be a group of order $p^k$ and $H \leq G$ a subgroup with…
Stabilizer commutes with conjugation II
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.2 Let $G$ be a permutation group on the set $A$ (i.e. $G \leq S_A$), let $\sigma…
Stabilizer commutes with conjugation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.1 Solution: First we prove that $\mathsf{stab}_G(b) = g \mathsf{stab}_G(a) g^{-1}$. ($\subseteq$) If $x \in \mathsf{stab}_G(b)$, then…
Conjugation by a fixed group element is an automorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.17 Let $G$ be a group and let $G$ act on itself by left conjugation; i.e., $g…
Conjugation is a group action
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.16 Let $G$ be a group. Show that the mapping defined by $g \cdot a = gag^{-1}$…
An alternate characterization of centralizer
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.1 Prove that $C_G(A) = \{ g \in G \ |\ g^{-1}ag = a\ \mathrm{for\ all}\ a…
Conjugation preserves order
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.22 Let $G$ be a group and let $x,g \in G$. Prove that $|x| = |g^{-1} x…