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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.]
Oct
31
comment Theorems with an extraordinary exception or a small number of sporadic exceptions
No free group is abelian, with the exception of $\mathbb Z$.
Oct
31
comment A problem dealing with Sylow's subgroups
@Galoisfan: You can even streamline my proof a little by unifying the argument that every element normalizes some Sylow subgroup.
Oct
31
answered A problem dealing with Sylow's subgroups
Oct
16
comment Is finite group theory still a fruitful area of research?
If you are interested, how a much improved proof of the classification of finite simple groups could look like, take a look at math.msu.edu/~meier/Preprints/CGP/cgp_abstract.html [This is a still active area, but technically quite difficult.]
Aug
30
comment A polynomial whose Galois group is $D_8$
Small typo: $F$-automorphism should be $\mathbb{Q}$-automorphism. As Dharam didn't specify the field (so I assume he doesn't care about it), I wonder why the solution on mathoverflow gets so many upvotes, but this generic solution didn't get any. Maybe you could mention what the fixed field for $G = S_n$ is, so that people recognize this solution from their algebra class...
Aug
23
comment Reference: Finite Groups and Geometry
The Geometry of the Classical Groups by D.E. Taylor.