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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".
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?
Do you just want to sketch the result or indeed the proof?
Sep
14
comment what groups have only elements of prime order?
The alternating group $A_5$ is a finite simple example.
Sep
12
comment $\operatorname{Aut}(G)$ contains an involution $\sigma$ with no nontrivial fixed point
It's not interesting because of its applications, but because of its generalization.
Jul
27
comment A group of order 2520
$G$ has a normal subgroup $N$ of order 21, which has a complement $H := C_G(K)$ of order 120. $H$ is a perfect central extension of $A_5$, whose center acts trivially on $N$ (by conjugation), so $A_5$ has to act on $N = C_7\rtimes C_3$. It fixes the characteristic subgroup $C_7$ of $N$, so it is not hard to see that $A_5$ acts trivially. In total you get that $G$ is the direct product of $N$ and $H$.
Jul
19
comment Historical Question about Schur-Zassenhaus Theorem
According to the articles about Hans Zassenhaus on wikipedia, his group theory book was based on lectures of his advisor Emil Artin and van der Waerden's algebra book was based on lectures by Artin and Emmy Noether.
Jul
17
comment Historical Question about Schur-Zassenhaus Theorem
Q1: According to the references in "The Theory of Finite Groups" by Kurzweil and Stellmacher Schur's paper should be "Schur, J.: Untersuchungen über die Darstellungen der endlichen Gruppen durch gebrochen lineare Substitutionen, J. reine u. angew. Math. 132 (1907), 85-137" as they usually cite the original papers (I didn't check). For the extended theorem by Zassenhaus they refer to the book.
Jul
17
comment Historical Question about Schur-Zassenhaus Theorem
Q3: The first proof of the conjecture was given by Feit and Thompson in their odd order theorem (no other proof was found yet).
Jul
8
comment Optimality proof for greedy algorithm
I just saw that I phrased my counterexample wrongly [I thought the gains of all steps add up]. If you don't have any other assumptions about the gains [to me it is not clear if in your model the order of actions matter, but this does not change much anyway], you could construct a example where greedy is not optimal, by assigning a "high value" x to some action $a$ and lower values to all other actions in the first step, and then by assigning values greater x to all pairs of actions (u, v) with $u\ne a$ and the value x to pairs of actions starting with a.
Jul
8
comment Optimality proof for greedy algorithm
Why do you expect the greedy algorithm to be optimal? Do you have any other (hidden) assumptions? [Otherwise take $\mathcal A = \{a, b, c\}$, $\mathcal B = 2$ (I assume this to be the number of actions before stopping, is this correct?), $g_a(S) = 3$, $g_b(S) = g_c(S) = 2$, $g_x(S|a) = 0$ for $x = a$ or $b$ and $g_x(S|y) = 2$ otherwise].
Jul
5
comment Definition question of convex orbit of finite group action
That the orbit is a convex subset.
Jun
28
comment Image of a normal Hall Subgroup under an automorphism
One problem in your reasoning is that you simply state "Further, let us define a subgroup H of order d."! How do you know that such a subgroup exists (it does)? You should either prove its existence or refer to some theorem that proves it [Or better: you don't need such a subgroup as Andreas showed].
Jun
26
comment If I know the Conjugacy classes of a group, do I know the group?
@jwodder: Take the product of Zev's example with a non-abelian group, e.g., $C_4\times S_3$ and $C_2^2\times S_3$.
Jun
25
comment How useful are geometric aspects when studying finite groups?
I can recommend reading 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).