# Wedderburn decomposition of $F(Z_2\times Z_2).$

Let $$F$$ be any finite field of characteristic different from $$2$$. I have to prove $$F(Z_2\times Z_2)\cong F\oplus F\oplus F \oplus F.$$ I know that $$F(Z_2\times Z_2)\cong F(Z_2)\otimes_F F(Z_2)$$ and $$F(Z_2)\cong F+F,$$ here $$\otimes$$ is the tensor product and $$F(G)$$ denotes the group algebra of the group $$G$$ over the field $$F$$. How to proceed further? Please help. Thanks.

Let's write the Klein 4-group multiplicatively as $$\mathbb Z_2\times\mathbb Z_2$$ as $$\{1,a,b,ab\}$$ with the usual rules regarding their multiplication (that is, they all square to $$1$$).

A useful fact (and easy exercise) is that if $$H$$ is a finite normal subgroup of a group $$G$$, and the order of $$H$$ is a unit in $$F$$, then $$\frac{1}{|H|}\sum_{h\in H}h$$ is a central idempotent of $$F(G)$$.

Let $$e=\frac{1+a}{2}$$, $$g=\frac{1+b}{2}$$ and $$f=\frac{1+a+b+ab}{4}$$.

These are all central idempotents according to the previous proposition.

Now one can check that $$\{f, e-f, g-f, 1-e-g+f\}=X$$ is a complete set of orthogonal idempotents for $$F(G)$$ in the sense that $$\sum_{x\in X}x=1$$ and $$xy=0$$ for all $$x,y\in X$$.

In such a case, $$F(G)\cong \prod_{x\in X}xF(g)x$$ as rings via the obvious map.

Maschke's theorem says that the ring is semisimple, and the Wedderburn theorem says $$F(G)$$ is a finite product of semisimple rings, but considering that $$F(G)$$ is commutative and we already have decomposed it into $$4$$ rings, the only possibility is that each of the rings is $$F$$.

• @rachwieb I am confused by the tensor product ... – neelkanth Jan 24 at 17:03
• How to handle tensor product in between ... – neelkanth Jan 24 at 17:04
• @neelkanth Yes, good question. I'm pretty sure we can get rid of it, I'm just trying to think of the easiest way to explain. Actually, I might fall back on applying my argument to the entire group, rather than focusing on the piece $F(\mathbb Z_2)$. – rschwieb Jan 24 at 17:06
• ok thanks ...... – neelkanth Jan 24 at 17:07
• @neelkanth I rewrote using a concrete argument. – rschwieb Jan 24 at 18:28