Intereting Posts

When does the kernel of a function equal the image?
If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$
Are $R=K/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K$ isomorphic?
A fast factorization method for Mersenne numbers
Notation for “the highest power of $p$ that divides $n$”
What is the prerequisite knowledge for learning Galois theory?
when does a separate-variable series solution exist for a PDE
Find the positive root of the equation $ce^{-c}-2(1-e^{-c})^2=0$
Prove Property of Doubling Measure on $\mathbb{R}$
Scalar triple product – why equivalent to determinant?
Proving that $f(x,y) = \frac{xy^2}{x^2 + y^2}$ with $f(0,0)=0$ is a continuous function using epsilon-delta.
Show that the Sorgenfrey line does not have a countable basis.
How to solve $x+\sin(x)=b$
What will be the one's digit of the remainder in: $\left|5555^{2222} + 2222^{5555}\right|\div 7=?$
Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$

I’m trying to solve exercise 8B.6 on page 249 of Isaacs’s *Finite Group Theory* textbook (the second question in a series; this is the third as question here). I have an idea, but it doesn’t quite work. It seems like the precursor to the idea in the next exercise, so I’d really like for it to work.

Can my idea be fixed?

The exercise asks:

- Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$?
- The comultiplication on $\mathbb{C} S_3$ for a matrix basis?
- How to determine the Galois group of irreducible polynomials of degree $3,4,5$
- Subgroups of $(\mathbb Z_n,+)$
- Order of the Rubik's cube group
- Extend isometry on some cube vertices to the entire cube

If $G$ is a finite primitive permutation group with a point stabilizer $H$ that has an orbit of size 2, then $G$ is dihedral of order twice an odd prime in its natural action.

The previous exercise had the stabilizer’s orbit size reduced to 1, and it followed by showing a maximal subgroup contained in two point stabilizers had to be the identity (which caused a huge collapse: $G$ is then cyclic of prime order). Ok, so I want to do the same thing, $H$ has a big subgroup contained in two stabilizers, so that big subgroup is the identity (which finishes the whole thing for my choice of subgroup).

Here was the smaller question I had while doing that, but I answered it depressingly as: “No.”

Suppose $H$ is a maximal subgroup of the finite group $G$, and let $K = H^2 = E^2(H)$ be the subgroup of $H$ generated by the squares of the elements in $H$. If $g\in G$ takes $K$ to $K^g ≤ H$, then must $g$ normalize $K$?

Well, taking $G$ to be $\operatorname{PGL}(2,q)$ for $q ≡ ±3\bmod 8$, and $H$ to be a Sylow 2-subgroup (so a dihedral group of order 8 with partial fusion), then $K$ is the order 2 normal subgroup of $H$ and is not weakly closed: there is a $g\in G$ that takes $K$ to a non-normal subgroup of order 2 in $H$. When $q = 3$, $H$ is a maximal subgroup of $G ≅ S_4$.

Now the action of $G$ on the cosets of $P$ is primitive, but not faithful, so this isn’t actually a counterexample, but wow it seems close.

Is it true that $K$ is weakly closed in $H$ (with respect to $G$) if $H$ is core-free (with respect to $G$)?

The only reason $H$ had a core was because $q$ was chosen so small. But all $q$ have the same basic setup as far as the relationship of $K$ in $H$ and conjugation in $G$, it is just that $H$ is no longer maximal for larger $q$.

- How to prove that if $G$ is a group with a subgroup $H$ of index $n$, then $G$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
- Determine the center of the dihedral group of order 12
- Isomorphism between the group $(\mathbb Z, +)$ and $(\mathbb Q_{>0}, .)$
- the representation of a free group
- Why is the group of units mod 8 isomorpic to the Klein 4 group?
- Is there a general formula for finding all subgroups of dihedral groups?
- let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)
- Does A5 have a subgroup of order 6?
- Drawing subgroup diagram of Dihedral group $D4$
- Isomorphisms between group of functions and $S_3$

- Why is the decimal representation of $\frac17$ “cyclical”?
- An inequality by Hardy
- Proving that $x^4 – 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.
- Describing the bended regions of a four-parameter logistic function
- Entire $f$ and $g$ constant if $e^{f(z)}+e^{g(z)}=1$
- What percentage of numbers is divisible by the set of twin primes?
- Some integral with sine
- How many different sizes of infinity are there?
- Minimal counterexamples of the isomorphism problem for integral group rings
- How can we factorise a general second degree expression?
- what type of math is this?
- Alternative definition of the determinant of a square matrix and its advantages?
- Maxima problem?
- “This statement is false” – Propositional Logic
- If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.