# Derek Holt

University of Warwick

warwick.ac.uk/~mareg

Age: 64

 16h comment Units in finite polynomial ringsIn particular, for $k=8$, $x^8+1=(x+1)^8$, and the units are the polynomials with an odd number of nonzero terms. 16h comment Units in finite polynomial ringsIn particular, for $k=8$, $x^8+1=(x+1)^8$, and the units are the polynomials with an odd number of nonzero terms. 22h comment Schur's theorem in transfer theoryGood, that's right! 22h comment Number of subgroups and elementsLet $K= \ker(\phi)$.Since $6$ divides ${\rm im}(\phi)$, you have $|K| = 5$ or $25$. It can't be $25$ by hypothesis, so $|K|=5$. Now $|G/K|=30$. Show that all groups of order 30 have a normal Sylow 5-subgroup, contradicting your hypothesis. 22h comment $F$ a finite field of $p^n$ elements. Suppose $F^\times=$. Then $\phi(x)=x^{p^r}$ for automorphisms$F$ is an extension of degree $p$ of the field ${\mathbb F}_p$ of order $p$ and there is a result in Galois theory that says that $F$ has a at most $n$ distinct automorphisms. But there are $n$ distinct automorphisms of the type you describe, so that must be the lot! 22h comment How many Groups there are on a finite countable set?But usually, different groups mean non-isomorphic groups. 22h answered schreier problem 1d comment Schur's theorem in transfer theoryI am afraid that it is not displaying very well on my computer, and I cannot see any underlined part. tell me where it begins and ends! 1d comment Schur's theorem in transfer theory$H=Z(G)$ and $M=1$. I am not sure what you mean by "how is the transfer map defined". You seem to know the definition of the transfer map. 2d comment Abstract algebra Centralizers / conjugacy classesIt is sufficient to conjugate $a$ by the elements in a set of right coset representatives of $C_G(a)$, so perhaps that is what was intended. Dec 3 comment Checking irreducibilityYes I meant easiest to implement. For groups that are not necessarily finite, I know nothing faster than the ideas suggested here by Peter and Yves. Dec 3 comment How to find the quotient group given a specific equivalence relation on $\mathbb{N}^3$You don't seem to have asked the right question. Why not edit it? Dec 3 comment Checking irreducibility@YvesCornulier: The method you describe is not used in practical algorithms for finite groups, because its complexity of $O(n^6)$ field operations is prohibitive, particualrly if you aim to do computations with $n \sim 10^5$. But it is probably the easiest for general f.g. groups provided that $n$ is not too big. Dec 3 comment Showing that if $N$ is minimal normal and $NH=G$, then $H$ is maximalSeems OK, but you should really consider the possibility $N \cap H = N$ (and similarly for $K$). That would imply $H=G$ contradicting $H < G$. Dec 3 comment How to find the quotient group given a specific equivalence relation on $\mathbb{N}^3$You need $(0,0,0)$ as an extra semigroup generator. I don't understand your last question - what do $(5,2)$,etc, mean here? Dec 3 comment How to find the quotient group given a specific equivalence relation on $\mathbb{N}^3$The image of $(0,1,0)$ is the equivalence class of $(0,1,0)$ under $\sim_L$, which contains the single element $(0,1,0)$. (But some equivalence classes have more than one element. For example the class of $(0,0,5)$ also contains $(3,4,0)$.) Dec 3 comment How to find the quotient group given a specific equivalence relation on $\mathbb{N}^3$If all you want to do is to find generators, then you can just use the images in $Q$ of generators of ${\mathbb N}^3$. By the way, there is no way of knowing whether or not you regard $0$ as an element of ${\mathbb N}$. Dec 3 comment Computation of large modulo valueYou type "36^333 mod 1333;" into an appropriate computer algebra system. (That's what I do anyway.) Dec 2 comment Determing the structure of the subgroup of an automorphism groupYou have it the wrong way round. You are trying to prove that $|{\mathbb Q}(t):{\mathbb Q}(u)|=2$ with $u=t(1-t)$, so you have to find the minimal polynomial for $t$ over ${\mathbb Q}(u)$. Dec 2 comment Determing the structure of the subgroup of an automorphism groupI don't understand what you are asking. Of course $t(1-t)$ is an element of ${\mathbb Q}(t)$. You were asked to identify a subfield of ${\mathbb Q}(t)$.