Brauer subsections and B-subsections

Assume the Axiom of Choice, and fix a splitting $p$-modular system for a finite group $G$. A Brauer subsection of $G$ is a pair $(u,c)$ in which $u\in G$ is a $p$-element and $c$ is a block of $k{C}_G(u)$. Subsections are considered up to simultaneous $G$-conjugacy: $(u,c)\sim (gu{g}^{-1},gc{g}^{-1})(g\in G)\mathrm{.}$ For a block $B$ of $kG$, the pair is a $B$-subsection when the induced block $c^G$ is $B$, in the local-to-global sense of A block induced from a subgroup. The associated $p$-section is $S_G(u)$ from The p-section of a p-element.

Let $H=C_G(u)$. The subgroup $⟨u⟩$ is a central $p$-subgroup of $H$, so every defect group $D$ of $c$ contains $⟨u⟩$ by Central p-subgroups lie in every block defect group. Hence $C_G(D)\le C_G(u)=H,$ and Centralizer containment makes block induction well-defined proves that $c^G$ is defined. Thus the notation does not silently assume the existence of an induced block.

Conjugation by $g$ identifies the block bimodule $c$ with the block bimodule $gc{g}^{-1}$ and carries every restriction summand in the definition of block induction to the corresponding conjugate summand. A global block ideal is fixed by inner conjugation because its idempotent is central. Therefore $(gc{g}^{-1}{)}^G=c^G,$ so the condition $c^G=B$ depends only on the subsection's conjugacy class. For $u=1$, the centralizer is $G$, the induced block is $c$ itself, and the definition reduces to the pairs $(1,B)$. The Axiom of Choice is used only to discharge the inherited published block-support and block-induction contracts; forming these finite conjugacy classes adds no choice (The Axiom of Choice).