Brauer correspondence

Let $B$ be a block of $kG$ with defect group $D$ and put $N=N_G(D)$. Its Brauer correspondent is the unique block $b$ of $kN$ with defect group $D$ selected by the nonzero Brauer image of the block idempotent of $B$.

Brauer's first main theorem gives a bijection between the blocks of $kG$ with defect group $D$ and the blocks of $k{N}_G(D)$ with defect group $D$. Corresponding blocks are related by their images under the Brauer morphism.

Over a splitting field of characteristic five, $A_5$ has a principal block of defect $C_5$ containing the ordinary characters of degrees $1,3,3,4$, and one defect-zero block containing the ordinary character of degree $5$. If $P$ is a Sylow 5-subgroup, then $N_{A_5}(P)\cong D_{10}$ and its unique 5-block is the Brauer correspondent of the principal block of $A_5$.