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Every doubly transitive group action is primitive

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.8 A transitive permutation group $G \leq S_A$ acting on $A$ is called doubly transitive if for…

Exhibit Dih(8) as a subgroup of Sym(4)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.11 Write out the cycle decomposition of the eight permutations in $S_4$ corresponding to the elements of…

Exhibit two subgroups which do not commute in Symmetric group S4

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.13 Fix any labeling of the vertices of a square and use this to identify $D_8$ as…

Compute in a group ring over dihedral group of order 8

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.9 Let $\alpha = r + r^2 - 2s$ and $\beta = -3r^2 + rs$ be elements…

Draw subgroup lattice of a quotient of quasi-dihedral group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.5 Let $QD_{16} = \langle \sigma, \tau \rangle$ be the quasidihedral group of order 16 described in…

Perform explicit computation in a quotient of a quasi-dihedral group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.18 Let $G = QD_{16}$ be the quasidihedral group presented by \langle \sigma, \tau \ |\ \sigma^8…

Perform computations in a quotient of dihedral group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17 Let $G = D_{16}$ and let $H = \langle r^4 \rangle$. (1) Show that the order…

Exhibit the cyclic subgroups of Dih(8) as sets

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.11 Find all cyclic subgroups of $D_8$. Exhibit a proper subgroup of $D_8$ which is not cyclic.…

Compute the center of Dih(2n)

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.7 Let $n \in \mathbb{Z}$ with $n \geq 3$. Prove the following. (1) $Z(D_{2n}) = 1$ if…

Demonstrate that a given subgroup is normal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.5 For each of the following subgroups $A$ of a given group $G$, show that \$C_G(A) =…