Featured on Meta Creating new Help Center documents for Review queues: Project overview thus the symmetric group behaves as though it were a linear group over "the field with one element". An algebra with 1 over a field, for example, contains a copy of the field, and the scalar action matches the ring action. $\endgroup$ – … Browse other questions tagged abstract-algebra commutative-algebra group-actions or ask your own question. When is there a difference between the group algebra over a field and an algebra over the same field, that is generated by a multiplicative subgroup that is isomorphic to the group? Algebras over a eld October 14, 2014 Roughly speaking, an algebra over a eld F is just a ring Rwith F contained in the center of R. In particular Ris an F-vector space, and this extra structure often simpli es life. It was first proved for finite p-groups by Tachikawa [Ta73Schu86 We study the group of invertible matrices over a finite field and consider an action of this group on a vector space and compute its stabilizer. For example, in the I think that for finite groups, the two notions should be $\begingroup$ Good answer: I would just add that that the ring of scalars is internal in many cases. I have definitely seen people struggle to find how some "algebra" was an "algebra over a field" when there was no connection other than the use of the word "algebra". The skew-field of quaternions is a $4$-dimensional algebra over the field of real numbers, while the Cayley–Dickson algebra is $8$-dimensional. It is known that NC-T2 holds for the group algebra kG of a finite group over a field k. This is a consequence of the work of Evens [Ev61] and Alperin and Evens [AlE81]. The dimension of any algebra with division over the field of real numbers is equal to 1, 2, 4, or 8 (see [Ad] , and also Topological ring ).

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