Intereting Posts

Solution to $e^{e^x}=x$ and other applications of iterated functions?
Constructing a quotient ring in GAP using structure constants
Prove that $a^2 + b^2 \geq 8$ if $ x^4 + ax^3 + 2x^2 + bx + 1 = 0 $ has at least one real root.
Show that $\dfrac{d}{dt}\left(ml^2\dot\theta\right)+mgl\sin\theta=l\dfrac{d^2\theta}{dl^2}+2\dfrac{d\theta}{dl}+\dfrac{g}{v^2}\theta$
An Ordinary Differential Equation with time varying coefficients
How to find the sum of this series : $1+\frac{1}{2}+ \frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{n}$
Question about permutation cycles
Definition of the image as coker of ker == ker of coker?
How is the acting of $H^{-1}$ on $H^1_0$ defined?
Does a monotone function on an arbitrary subset of $\mathbb R$ always have at most countable number of discontinuity?
The Leibniz rule for the curl of the product of a scalar field and a vector field
Quintic diophantine equation
Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?
Why is the category of groups not closed, or enriched over itself?
How to show that the commutator subgroup is a normal subgroup

This is exercise 3.2.24 from Scott, *Group Theory*.

If $H$ is a finite maximal abelian normal subgroup of $G$ and $K$ is a normal abelian subgroup of $G$, then $K$ is finite.

The hint is to use Normalizer/Centralizer theorem.

- For finite abelian groups, show that $G \times G \cong H \times H$ implies $G \cong H$
- For abelian groups: does knowing $\text{Hom}(X,Z)$ for all $Z$ suffice to determine $X$?
- How to prove that homomorphism from field to ring is injective or zero?
- Are there any distinct finite simple groups with the same order?
- What can we say about the kernel of $\phi: F_n \rightarrow S_k$
- If G is not commutative, then is there always a subgroup that is not a normal subgroup?

- f(a) = inverse of a is an isomorphism iff a group G is Abelian
- Complex numbers modulo integers
- If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
- how many element of order 2 and 5 are there
- Divisibility question
- Adapting a proof on elements of order 2: from finite groups to infinite groups
- Painting the faces of a cube with distinct colours
- $ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
- Square free finite abelian group is cyclic
- Are cyclic groups always abelian?

Since $H$ is finite, $Aut(H)$ is finite. By the Normalizer/Centralizer theorem, $\frac{N_{G}(H)}{C_{G}(H) }= \frac{G}{C_{G}(H)} \ $ is isomorphic to a subgroup of $Aut(H)$ and so is finite. Now we note that if $M \ \trianglelefteq \ G \ $, $M$ is abelian and $M \leq C_{G}(H) \ $, then $HM$ is abelian and normal in $G$, but $H$ is maximal abelian normal so $HM\leq H \ $ and then $M\leq H \ $. Note that

$K \cap C_{G}(H) \ $ is abelian because $K$ is abelian and $K \cap C_{G}(H) \trianglelefteq G \ $ because $K \trianglelefteq G \ $ and $C_{G}(H) \trianglelefteq G$. Then $K \cap C_{G}(H) \subseteq H \ $ and so $K \cap C_{G}(H) \ $ is finite because $H$ is finite. $\frac{KC_{G}(H)}{C_{G}(H)} \simeq \frac{K}{K \cap C_{G}(H)} \ $ is a subgroup of $\frac{G}{C_{G}(H)} \ $ and so is finite. Then $|K| = |\frac{K}{K \cap C_{G}(H)}| \cdot |K \cap C_{G}(H)| \ $ is finite.

- Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$
- Solving a double integration in parametric form
- How many ways to divide group of 12 people into 2 groups of 3 people and 3 groups of 2 people?
- How many words can be formed from 'alpha'?
- Solve by induction: $n!>(n/e)^n$
- Unit ball in $C$ not sequentially compact
- Classifying Types of Paradoxes: Liar's Paradox, Et Alia
- Multiplication of a random variable with constant
- Dimension of the solution of a second order homogenous ODE
- How discontinuous can a derivative be?
- Polynomial with integer coefficients ($f(x)=ax^3+bx^2+cx+d$) with odd $ad$ and even $bc$ implies not all rational roots
- Tangent bundle of P^n and Euler exact sequence
- Example of strictly subadditive lebesgue outer measure
- Solving for $a$ in power tower equation
- A field that is an ordered field in two distinct ways