## The Group Algebra (Pt. II)

**Point of post: **This is a continuation of this post.

Our next theorem is more of an observation but is worth mentioning

**Theorem: ***Let be a finite group and the group algebra over . Then, is a basis for .*

**Proof: **It’s evident that is linearly independent since if

implies in particular that

for each from where linear independence follows. The fact that follows immediately from the evident formula

for any . The conclusion follows.

**Corollary: ***Let be a finite group and the group algebra over . Then, .*

Our next theorem just has to do with a couple of the menial, but convenient algebraic properties of the group algebra. In particular:

**Theorem: ***Let be a finite group and the group algebra over . Then, **the following properties hold for all , , and :*

**Proof:**

: We merely note that for any we have that

and that this sum will be zero if there does not exists a such that and and if it does. In other words, the sum will be zero if and if . But, this implies that for any We have that from where it follows that .

: We merely note for any we have

and since was arbitrary the conclusion follows.

: This is fairly clear since for any we have that

since was arbitrary the conclusion follows.

: This is also fairly clear since for any we have that

since was arbitrary the conclusion follows.

: Lastly we merely note that for any we have that

since was arbitrary the conclusion follows.

**References:**

1.Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Prin

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