Are there finite-dimentional unital associative algebras over $\Bbb{C}$ that are not isomorphic to a group algebra $\Bbb{C}[G]$ for finite group G?

Question

I've seen somewhere in my lecture notes that the answer is yes, but I can't think of an explicit example of such an algebra.
Is there a standard way of constructing these algebras for particular groups G?

Answer 1

Note that since any group $G$ has the trivial rep'n, the group ring $\mathbb C[G]$ always contains a copy of $\mathbb C$ as a direct factor. So any $\mathbb C$-algebra without this property gives an example, e.g. $M_n(\mathbb C)$ for any $n > 1$.

There are other fairly obvious constraints that group rings satisfy (related to the facts that dim $V | | G||$ for each irrep. $V$, and that the sums of the squares of the dimensions of the non-isomorphic irreps. equals $|G|$). But the one I used above is perhaps the simplest in terms of constructing such (counter)examples.

Answer 2

Just to consider another example, any non-commutative algebra of prime dimension qualifies, as groups of prime order are abelian.

Answer 3

By Maschke's theorem $\mathbb{C}[G]$ is semisimple, so by the Artin-Wedderburn theorem it is isomorphic to a direct product of matrix algebras $M_d(\mathbb{C})$, one for each $d$-dimensional irreducible representation of $G$. This identifies lots of constraints on $\mathbb{C}[G]$ as an algebra (several of which have already been identified in other answers):

• First, it must be semisimple. This is a very strong condition. Most algebras are not semisimple; the smallest example is $\mathbb{C}[x]/x^2$, which has dimension $2$.
• Second, as Matt E points out, it must contain $\mathbb{C}$ as a factor. Many algebras, even semisimple algebras, do not contain $\mathbb{C}$ as a factor: the smallest semisimple example is $M_2(\mathbb{C})$, which has dimension $4$.
• Third, also as Matt E points out, by character theory the dimensions $d$ of the irreducible representations must divide the dimension $|G| = \sum d^2$ of the whole algebra. Many semisimple algebras don't have this property; the smallest example is $\mathbb{C} \times M_2(\mathbb{C})$, which has dimension $5$.
• Fourth, as Andreas Caranti and others point out, in some cases you know every group of order $|G|$ and so can rule out most semisimple examples of the same dimension. For example, whenever $|G|$ is prime, the unique corresponding group is abelian and hence only has $1$-dimensional representations, so all other semisimple algebras of prime dimension are ruled out. The smallest example is again $\mathbb{C} \times M_2(\mathbb{C})$, which has dimension $5$, and the next smallest example is $\mathbb{C}^3 \times M_2(\mathbb{C})$, which has dimension $7$.