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May
20
comment A question from Isaac's Book
@JackSchmidt: Is your last comment a statement or a question? (The exercise is 8.2.11 in Kurzweil/Stellmacher - and they need solvability of either $G$ or $A$.)
May
15
comment Is group theory useful in any way to optimization?
Please be so polite to give at least links to the other place when posting twice the same question.
May
15
comment Proving facts about groups with representation theory.
@TobiasKildetoft: Purely group-theoretic proofs of Burnside's theorem were given by Goldschmidt (1970, odd), Bender (1972, general) and Matsuyama (1973, even). But Bender provided also a purely group-theoretic proof for Frobenius groups with $H$ of even order.
Jan
31
comment Proving facts about groups with representation theory.
H. Bender found a very simple, purely group-theoretic proof for 2. in case that $H$ as even order. A Fourier-analytic proof can be found here.
Nov
13
comment On direct product
Which direction of the proof do you have problems with?
Nov
7
comment Bijection map from a set of subgroup to another set of subgroup under some condition.
@user1729: Ah thanks, on my other computer I see the sign for normality double and thought $N$ is only subnormal...
Nov
6
comment How to show that $S(n, q)$ is a chamber complex?
For $n = 2$ the vertices of $S(2, q)$ are all $1$-dimensional, so none can contain another one. Walls are empty sets. So the answer to the question at the end is "No". (A good source for a proof of your first question is the book The Geometry of the Classical Groups by D.E. Taylor.)
Nov
6
comment Bijection map from a set of subgroup to another set of subgroup under some condition.
In my addition of Kurzweil/Stellmacher this exercise is given at the end of section 1.3 with $N$ normal in $G$. Which section is your exercise from?
Oct
29
comment Why is conjugation by an odd permutation in $S_n$ not an inner automorphism on $A_n$?
You could look at the kernel of the map $S_n \to \mathrm{Aut}(A_n)$ given by conjugation.
Oct
24
comment Group of order 24 with no element of order 6 is isomorphic to $S_4$
Building on Tobias' comment: The action of $G$ is transitive, so the kernel has order dividing $6$. A $3$-subgroup fixing a $3$-Sylow (with respect to the action by conjugation) is contained in it (as their product is a $3$-group), so a $3$-Sylow of the kernel is trivial. As a normal subgroup of order $2$ is central, we get that the action is faithful. q.e.d.
Oct
10
comment Is there a geometric idea behind Sylow's theorems?
For connections between Sylow's theorems and geometry take also a look at this answer by Vipul Naik to a similar question or at the article Subgroup complexes by Peter Webb, pp. 349-365 in: ed. P. Fong, The Arcata Conference on Representations of Finite Groups, AMS Proceedings of Symposia in Pure Mathematics 47 (1987).
Oct
10
comment Is there a geometric idea behind Sylow's theorems?
Looking at the action on the set of maximal $p$-subgroups (i.e., maximal elements in the set of $p$-subgroups) by conjugation leads also quickly to Sylow's theorems, but requires Cauchy's theorem as starting point.
Oct
3
comment Is there a non-trivial binary operation on the set of subgroups of a finite group that distributes with intersection?
What do you get if you use this three subgroups of $S_3$ in your attempt (where $L$ is called $G$)?
Oct
2
comment Is there a non-trivial binary operation on the set of subgroups of a finite group that distributes with intersection?
Try $H, K, L$ the three different subgroups of order $2$ of symmetric group $S_3$ on three elements.
Oct
2
comment Show that there exists two Sylow $p$-subgroups $P$ and $Q$ such that $[P:P\cap Q] = [Q:P\cap Q] = p$
Which are the orders $\bmod p^2$ of the orbits other than $\{P\}$?
Oct
1
comment How many subgroups does $\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 $ have?
@TobiasKildetoft: I just wanted to get rid off of the unnecessary word "abelian".
Oct
1
comment Missing $\{2,p\}$-Hall subgroups in finite non-abelian simple groups
Did you see the related (but different) question mathoverflow.net/questions/119220/… ?
Oct
1
comment How many subgroups does $\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 $ have?
Where do you need that $G_1$ and $G_2$ are abelian in your proof?
Sep
30
comment How many subgroups does $\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 \times\mathbb{Z}_6 $ have?
Hint: $Z_6 = Z_2\times Z_3$ implies $H = Z_2^4\times Z_3^4$. The subgroups of $Z_2^4$ and $Z_3^4$ you should be able to determine. As $Z_2^4$ and $Z_3^4$ have coprime order, what can you say for $U\le H$ about $(U\cap Z_2^4)\times (U\cap Z_3^4)$?
Sep
15
comment How can I explain the sketch of the proof of classification of finite simple groups to someone who knows only basics of algebra?
You can take a look either at www.ams.org/notices/199502/solomon.pdf or at the first volume of "The Classification of the Finite Simple Groups" by Daniel Gorenstein, Richard Lyons and Ronald Solomon (which - if I remember correctly - contains a sketch of the proof). For a sketch of the proof you might not need to explain how to construct the families of groups of Lie type, as in the proof all work is "done in characteristic 2".