Finite vs Ring Groups: Examining Theorems

[cbarker1]

Dear Everyone,

Does every theorem that holds for finite group holds for ring groups? Why or Why not?


Thanks
Cbarker1

 

[Olinguito]

What do you mean by “ring groups”?

 

[cbarker1]

A group ring defined as the following from Dummit and Foote:

Fix a commutative ring $R$ with identity $1\ne0$ and let $G=\{g_{1},g_{2},g_{3},...,g_{n}\}$ be any finite group with group operation written multiplicatively. A group ring, $RG$, of $G$ with coefficients in $R$ to be the set of all formal sum

$a_1g_1+a_2g_2+\cdots+a_ng_n$, $a_i\in R$, $1\le i\le n$

 

[Olinguito]

Well, note that $G$ is a group whereas $RG$ is a ring, so not every theorem about $G$ may be applicable to $RG$. For example, $G$ may be a cyclic group, but there is no such thing as a cyclic ring, so a theorem about cyclic groups may not make sense when applied to rings.

What you can say is that $RG$ contains a subring isomorphic to $R$, namely
$$\{a\cdot e_G:a\in R\}$$
as well as a subset which, with respect to multiplication, forms a group isomorphic to $G$, namely
$$\{1_R\cdot g:g\in G\}.$$