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This is contained in Jacobson's Blue Book (Structure and Representations of Jordan Algebras, AMS Colloquium Publications, 1968), as Exercise 1 on p. 342, for arbitrary fields, and with no assumptions …

My answer is definitely less complete than that of Mark Sapir, but in case you want to see an explicit example: the easiest counterexample is the dihedral group $D_{12}$, which you can view either as …

This is contained in section 1.5.3 of the book "Dualities and representations of Lie superalgebras" by Cheng and Wang. Chapter 1 of their book happens to be available for free on the AMS bookstore web …

A classification is too much to hope for, but the representation theory tells you whether such an algebra structure can exist: if $V$ is your $G$-representation, then an algebra product corresponds to …

I think that Waterhouse's "Introduction to Affine Group Schemes" (1979), section 3.2 "Comodules", might be what you're looking for. I'm not sure why you have the constraint "non-zero characteristic" …

I might be missing something, but it seems to me that there is not much going on in your construction. In fact, your original action of $Z_n$ on $X$ does nothing more than putting a cyclic ordering on …

I believe there is a good reason why mathematicians don't use the terminology "simplex-part" and "perplex-part": they are not canonical! Indeed, algebraically there is no way to distinguish the elemen …

An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here). …

There is indeed a strong analogy between the study of $p$-local subgroups and the theory of buildings, at least for groups of Lie type. More precisely, if $G$ is a finite group of Lie type over a fiel …

If $k$ is a commutative field of characteristic $p>0$, then the map $$ \theta \colon \mathrm{GL}_p(k) \to \mathrm{SL}_p(k) \colon A = (a_{ij}) \mapsto (\det A)^{-1} (a_{ij}^p) $$ is a group homomorphi …