All Questions
Tagged with finite-fields finite-groups
57 questions
11
votes
3
answers
1k
views
Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?
Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...
- 375
10
votes
8
answers
1k
views
Classifications of finite simple objects
I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "...
15
votes
1
answer
4k
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Order of finite unitary group
This may be an easy exercise but I am not getting it. Let $\mathbf F_q$ be a finite field with $q$ elements and $\mathbf F_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of $\...
- 899
4
votes
0
answers
1k
views
Representations of general linear groups GL_n(F_q) - decomposition of tensor product?
Let $V$ and $W$ be complex irreducible representations of $GL_n(F_q)$ where $F_q$ is finite field. Is the decomposition of $V \otimes W$ into irreducible representations known?
PS
Same question:
...
- 899
19
votes
4
answers
2k
views
Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?
A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...
- 9,389
3
votes
2
answers
1k
views
Cyclic order relation in Zn
The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?
- 1,190
32
votes
9
answers
5k
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How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?
Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
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- 236k