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2d
comment The Weyl group of E8 versus $O_8^+(2)$
The Atlas gives the order of $O_{8}^{+}(2)$ as 174182400.
2d
comment The Weyl group of E8 versus $O_8^+(2)$
Things are also complicated by the fact that the simple group $O_{8}^{+}(2)$ has a Klein $4$-group as its Schur multiplier.
2d
comment The Weyl group of E8 versus $O_8^+(2)$
Mmm: according to the Atlas, the Weyl group of $E_{8}$ has structure $2.G.2$, where $G$ is the simple group $O_{8}^{+}(2)$. This may be due to different naming conventions for orthogonal groups. Recall that the special orthogonal group can have a simple subgroup of index $2$ in some cases.
2d
comment How to calculate the derivative of logarithm of a matrix?
One thing to note is that while the series for $\exp(M)$ converges for any complex square matrix $M$, the series you give for $\log(M)$ may not converge if $\|I -M \| > 1$.
Jan
29
comment Uniqueness of the fusion ring for simple finite group
@QiaochuYuan : OK, no problem.
Jan
29
comment Uniqueness of the fusion ring for simple finite group
@Qiaochu Yuan: My last comment was to the second comment above, which seems quite clear to me: he seems to be asking whether $A_{c}(G)/{\rm Inn}(G)$ is non-trivial, where $A_{c}(G)$ is the subgroup of automorphisms of $G$ which fix all irreducible characters (at least that is what the question in comments say). The answer to that question is "no", in fact that quotient group need not be Abelian. In a similar vein, to clarify : Huppert's conjecture is different from asking whether the character table determines a simple group (which as you correctly point out in your answer, it does).
Jan
29
comment actions of the hyperoctahedral group
I'm sure it must, since it is natural, but I know no reference.
Jan
29
comment actions of the hyperoctahedral group
You just take an $n$-dimensional (linear) representation of $H_{n}$ as the group of all monomial matrices with non-zero entries $\pm 1$. This may be viewed as a linear representation over any field of odd characteristic. So for $p$ an odd prime, this gives a permutation action of $H_{n}$ on the $p^{n}-1$ non-zero vectors in an $n$-dimensional vector space over the field of $p$ elements. Only the identity of $H_{n}$ fixes all vectors .
Jan
29
answered actions of the hyperoctahedral group
Jan
29
comment Uniqueness of the fusion ring for simple finite group
That is definitely not true for general finite groups unfortunately. I think Burnside already knew this. I think C.H. Sah constructed examples where the quotient of that subgroup of Aut(G) by Inn(G) is not even Abelian.
Jan
29
comment Uniqueness of the fusion ring for simple finite group
It is a (consequence of a) conjecture of B. Huppert that two non-Abelian finite simple groups which have the same set of complex irreducible character degrees (even ignoring multiplicities) should be isomorphic. I am not sure how widely this has been checked to date though some cases are known.
Jan
28
answered How many solutions does $\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}=1$ have?
Jan
28
answered How do I determine a real matrix form for a group representation?
Jan
25
comment Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?
Yes,thanks. I saw Igor Rivin's answer too.
Jan
25
comment A group algebra isomorphism problem
(Corrected): I suppose you could simplify the statement a bit: by stating that the field should have n distinct roots of unity whenever $G$ has an element of order $n$, you could omit the statement about the characteristic ( which would be implied by the other conditions).
Jan
25
comment A group algebra isomorphism problem
Oops yes I didn't say what I meant. I'll rewrite the comment, It' only a minor point anyway, but might as well say it right.
Jan
24
comment the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group
It depends how pedantic you want to be. Strictly speaking, yes, you need to know that the vector space you want to prove has certain properties is actually a vector space in the first place, so is non-empty in particular.
Jan
24
comment the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group
Well, it is nonzero because there is the trivial representation (which sends every element of $G$ to $1 \in \mathbb{C})$).
Jan
23
comment Question related to Fermat curve: Does the equation $A x^n + By^n = C z^n$ have any solution in $\mathbb{N}$?
You need to exclude some other degenerate cases to make the question interesting: for example, if $x = y = z$ and $A+B = C$, there will be a solution.
Jan
23
comment Does the antidiagonal in this square matrix always contain a prime?
I revised some earlier comments I made in light of the reaction they got.I would like to see how the numbers work.