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seen Sep 24 at 10:54

Mar
21
comment Are there/Why aren't there any simple groups with orders like this?
@caveman: I think it would be nice if you could add your formulas to the question.
Mar
21
comment Are there/Why aren't there any simple groups with orders like this?
@caveman: What exactly did you mean with your remark "ignoring the matrix groups for which the problem is solved"?
Mar
21
comment Are there/Why aren't there any simple groups with orders like this?
@SteveD: The Lie-type groups are probably excluded by the remark "ignoring the matrix groups for which the problem is solved". (I think, the exponent of $p$ in $|PSL_n(p)|$ grows quadratically for $n \to \infty$, but only linear for all other primes.)
Feb
27
comment Normal subgroups of finite index in infinite direct sum
Calling $\mathop{Supp} x = \{i\in I: x_i \ne 1\}$ the support of $x \in$ your infinite product/sum, take a look at the set $N_U$of all $x$ whose support is not an element of some fixed non-principal ultrafilter U on I. It's not too hard to show that $N_U$ is a normal subgroup. If all $S_i$ are isomorphic to each other, I think its index should be finite.
Feb
18
comment How do I understand $e^i$ which is so common?
Take also a look at this answer to a related question at mathoverflow.
Feb
6
comment What is a minimal polynomial of a group element, and why would we care if it was quadratic?
If you can get the book The Theory of Finite Groups by Kurzweil and Stellmacher, take a look at chapter 9 "Quadratic Action".
Jan
31
comment References about finite group theory
*The Theory of Finite Groups" by Kurzweil and Stellmacher is not as comprehensive as Aschbacher's book and its first half is a good introduction for beginners, but its second half (Quadratic Action, The Embedding of $p$-Local Subgroups, Signalizer Functors and N-Groups) goes quite deep and shows also in some special case how to classify simple groups that are $N$-groups, i.e., groups of even order with all $2$-local subgroups solvable [usually $N$-group is the stronger condition that all $p$-local subgroups are solvable].
Jan
19
comment A few questions about nonabelian cohomology of finite groups.
You could take a look at the chapter "The Theory of Group Extensions" in Derek Robinson's book A Course in the Theory of Groups. One section is called "Group-Theoretic Interpretations of the (Co)homology groups".
Jan
16
comment Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
@DonAntonio: Are you sure about your statement for non-abelian groups of order $p^3$? $Q_8$ has the property, but a non-abelian group of order $p^3$ and exponent $p$ (implying $p>2$) doesn't have it.
Jan
13
comment On inner automorphisms group of semidirect product of two groups
You are right about doubting the inclusion. As $\mathop{Inn}(G) = G/\mathop{Z}(G)$ you will have to determine (I change notation!) the center of $G = N\rtimes H$, which is generated by the central elements of $N$ that are centralized by $H$ and by the central elements of $H$ that centralize $N$. An example where the inclusion becomes nonsense is $N = Z_3^2$ and $H = Q_8$ with the faithful action of the quaternion group.
Jan
12
comment List of finite groups of Lie type and their BN-pairs
Take a look at "The Geometry of Classical Groups" by Donald E. Taylor. It's quite readable, and you can find descriptions of the BN-pairs for all classical groups there.
Dec
24
comment Prime divisor in the Automorphism group
Take a look at the Suzuki groups, according to wikipedia they can have automorphisms of order $r = 3$.
Dec
24
comment Prime divisor in the Automorphism group
Do you want $G$ to be non-abelian?
Nov
25
comment Maximal soluble subgroups in a parabolic subgroup of finite classical simple group
If you ask the same question here and on mathoverflow, please give a link to the other page to avoid duplicated work by people not checking both pages!
Nov
15
comment Isomorphism between centralizer and semi direct product
Hint: Let $\sigma\in S_{mb}$ be your product of $m$ disjoint $b$-cycles. As a first step you should be able to show that elements centralizing $\sigma$ permute the orbits of $\langle\sigma\rangle$.
Nov
9
comment If a finite group has $\geq p+1$ Sylow $p$-subgroups (of order $p^n$), then there are $\geq p^{n+1}$ elements in the Sylow $p$-subgroups
@JackSchmidt: They are all in Sylows. If they are not in distinct Sylows, the intersection is not maximal.
Nov
9
comment If a finite group has $\geq p+1$ Sylow $p$-subgroups (of order $p^n$), then there are $\geq p^{n+1}$ elements in the Sylow $p$-subgroups
@DerekHolt: Yes, this was the idea.
Nov
9
comment If a finite group has $\geq p+1$ Sylow $p$-subgroups (of order $p^n$), then there are $\geq p^{n+1}$ elements in the Sylow $p$-subgroups
Hint: Take $Q$ be maximal among the intersections of $p$-Sylow subgroups. How many $p$-Sylow subgroups contain $Q$, and what do you know about their intersections? [Apply Sylow's theorem to $N_G(Q)$.]
Nov
7
comment Number of solutions of $x^d = 1_G$ .
See also mathoverflow.net/questions/109027/…
Nov
7
comment Prove that there are no simple groups of order 224.
There are even less $2$-Sylows ($G \to S_7$). [I guess you mean injective in the last sentence.]