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…
Subgroups of finite index force the existence of normal subgroups of bounded index
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.8 Solution: $G$ acts on the cosets $G/H$ by left multiplication. Let $\lambda : G \rightarrow S_{G/H}$…
Basic properties of blocks of a group action
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.7 Let $G \leq S_A$ act transitively on the set $A$. A block is a nonempty subset…
Compute the orbits, cycle decompositions, and stabilizers of some given group actions of Sym(3)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.4 Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq…
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…
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…
The kernel and stabilizers of a group action are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.4 Let $G$ be a group acting on a set $A$. Show that the following sets are…
Compute the stabilizer of an element under a given action of Sym(n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Fix $i \in \{ 1, \ldots, n \} = A$, and let $S_n$ act on $A$…
Sym(4) acts on a set of polynomials by permuting the variables
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.12 Let $R$ be the set of all polynomials with integers coefficients in the independent variables $x_1,…
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