Intereting Posts

The commutator subgroup of a quotient in terms of the commutator subgroup and the kernel
Bounded convergence theorem
Surjective endomorphisms of finitely generated modules are isomorphisms
How to show that a continuous map on a compact metric space must fix some non-empty set.
Interesting puzzle about a sphere and some circles
Galois extension of $\mathbb{Q}$ with Galois group $\mathbb{Z}/4\mathbb{Z}$ that contains $\mathbb{Q}(\sqrt{3})$?
When does Schwarz inequality become an equality?
Strictly diagonally dominant matrices are non singular
Are all algebraic integers with absolute value 1 roots of unity?
Publishing elementary proofs of theorems
What's the smallest number that we can multiply with a given one to get the result only zeros and ones?
If $A=AB-BA$, is $A$ nilpotent?
First proof of Poincaré Lemma
Functional equation book for olympiad
Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

A group of order $3^a\cdot 5\cdot 11$ has a normal Sylow $3$-subgroup.

This is question 5C.7 in Isaacs’s *Finite Group Theory*.

That section of the text is about transfer in finite groups, and proves results like Burnside’s transfer lemma, that groups with all Sylows cyclic are metacyclic, etc. As for the problem, if $|G|=3^a\cdot5\cdot11$, I can, after checking a few cases and using induction, reduce to the case that $G$ is simple. But I don’t see how to show, using what has come before, that $G$ can’t be simple.

- Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.
- Find a $2$-Sylow subgroup of $\mathrm{GL}_3(F_7)$
- Infinitely many simple groups with conditions on order?
- Given a finite Group G, with A, B subgroups prove the order of AB
- Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.
- Existence of subgroup of order six in $A_4$

And actually, my bigger problem is that it seems like *a lot of work* for one problem. And even more problematic is the fact I haven’t really used any results related to transfer! Is there another, easier route? Or perhaps is there a missing “abelian” in the problem?

Thanks!

- Direct product of two nilpotent groups is nilpotent and direct product of two solvable groups is solvable
- Isomorphism between groups of real numbers
- What's an easy way of proving a subgroup is normal?
- If a finite group $G$ is solvable, is $$ nilpotent?
- How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?
- why is a polycyclic group that is residually finite p-group nilpotent?
- Does $|x^G|$ divide the order $\langle x^G\rangle$?
- Precise connection between complexification of $\mathfrak{su}(2)$, $\mathfrak{so}(1,3)$ and $\mathfrak{sl}(2, \mathbb{C})$
- If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
- $G$ finite group, $H\leq G$ such that $C_G(x)\subseteq H\quad\forall x\in H$ such that $p\mid o(x)$

As an alternative to your proof (given in a comment, already simplified by Geoff) you could also just use Sylow’s theorem and no transfer (surely not intended by Isaacs):

By Sylow the number of $3$-Sylow subgroups has to be either $1$ or $55$. In the first case, you are done, so assume the latter case. Choose two $3$-Sylow subgroups $S\ne T$ such that their intersection $P = S\cap T$ is maximal among intersections of Sylow subgroups.

As $P < \mathrm{N}_S(P)$ and $P < \mathrm{N}_T(P)$, you get by the maximal choice of $P$ that the normalizer $H = \mathrm{N}_G(P)$ does not have a unique normal $3$-Sylow subgroup. As the order of $H$ is $55$ times a power of $3$, $H$ has as many $3$-Sylow subgroups as $G$, each of them properly containing $P$. As $P$ is a maximal intersection of $3$-Sylows of $G$, each $3$-Sylow subgroup of $H$ is contained in a unique $3$-Sylow subgroup of $G$, showing that $P$ is also contained in *every* $3$-Sylow subgroup of $G$. Hence $P$ is the intersection of all $3$-Sylow subgroups of $G$ and therefore normal in $G$.

Now look at $\bar{G} = G/P$: $\bar{G}$ has $55$ $3$-Sylow subgroups which intersect pairwise trivially (again by maximal choice of $P$) and are self-normalizing (i.e., $\mathrm{N}_{\bar{G}}(\bar{S}) = \bar{S}$). Therefore $\bar{G}$ contains exactly $54$ elements whose order are not a power of $3$.

By Sylow $\bar{G}$ has a unique $11$-Sylow subgroups (there are not enough $3'$-elements for more $11$-Sylow subgroups than that), which is normal and centralized by every non-trivial $3$-element as $\mathrm{Aut}(C_{11})$ is cyclic of order $10$, contradicting that the $3$-Sylow subgroups of $\bar{G}$ are self-normalizing.

- The security guard problem
- Explanation on arg min
- Primitive roots modulo primes congruent to n!
- Are two subgroups of a finite $p$-group $G$, of the same order, isomorphic?
- About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
- Prove that 3 is a primitive root of $7^k$ for all $k \ge 1$
- How does Lambert's W behave near ∞?
- How to manually calculate Roots
- The minimum perimeter and maximum height of a triangle under constraints
- $x^p -x-c$ is irreducible over a field of characteristic $p$ if it has no root in the field
- Estimate a sum of products
- Groups of order 42 and classification.
- Are there arbitrarily large sets $S$ of natural integers such that the difference of each pair is their GCD?
- How can I define a topology on the empty set?
- Solution to $x^n=a \pmod p$ where $p$ is a prime