n divides the totient of $p^n-1$ when p is prime
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.15 Let $p$ be a prime and let $n$ be a positive integer. Find the order of…
Fermat’s Little Theorem
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.16 Use Lagrange’s Theorem in the multiplicative group $(\mathbb{Z}/(p))^\times$ to prove Fermat’s Little Theorem: if $p$ is…
Solution to Mathematics for Machine Learning Exercise 7.6
Consider the linear program illustrated in Figure 7.9, $$ \min _{\boldsymbol{x} \in \mathbb{R}^{2}}\quad -\left[\begin{array}{l} 5 \\ 3 \end{array}\right]^{\top}\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right] $$ $$ \text { subject to }\quad \left[\begin{array}{cc}…
Solution to Mathematics for Machine Learning Exercise 7.2
Consider the update equation for stochastic gradient descent (Equation (7.15)). Write down the update when we use a mini-batch size of one. Solution: Let $a$ be any function from $\mathbb…
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) has no normal subgroups of order 8 or 3
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.14 Prove that $S_4$ does not have a normal subgroup of order 8 or a normal subgroup…
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…
Given a subgroup of a group, the numbers of left and right cosets are equal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.12 Let $G$ be a group and $H \leq G$. Prove that the map $x \mapsto x^{-1}$…
Subgroup index is multiplicative across intermediate subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.11 Solution: See Lemma 3 of Exercise 3.2.10.
Bounds on the index of an intersection of two subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.10 Let $G$ be a group and let $H,K \leq G$ be subgroups of finite index; say…