# How exactly does character inflation work?

Let $$A_4$$ be the alternating group and let $$V = {\{(1), (12)(34), (13)(24), (14)(23)}\}$$ be a normal subgroup of $$A_4$$. Then $$A_4/V \simeq C_3$$, so $$A_4$$ has $$3$$ one dimensional representations, which apparently, we can inflate/lift from the character table for $$C_3$$.

I don't really understand how this works? How do I relate the character values of $$C_3$$ to $$A_4$$? Do I need to define a homomorphism from $$A_4$$ to $$C_3$$?

• Just look at the representation $A_4 \to A_4 /V \to GL(X)$ induced from a representation $A_4/V \to GL(X)$. – anomaly Jan 21 at 22:00