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Apr
13
comment $p$-Group as union of subgroups
Do you happen to know the Frattini subgroup (the intersection of all maximal subgroups)? If you look at the group modulo its Frattini subgroup everything becomes more easy.
Apr
13
comment $p$-Group as union of subgroups
What did you intend to say with your first sentence "It is well known that a group can not be union of proper subgroups."? As every group is the union of its cyclic subgroups your statement holds only for cyclic groups. Did you want to write "of two proper subgroups"?
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
23
awarded  Yearling
Feb
21
answered What is a minimal polynomial of a group element, and why would we care if it was quadratic?
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.