Abstract
We discuss representations of finite groups having a common central 𝑝-subgroup 𝑍, where 𝑝 is a prime number. For the principal 𝑝-blocks, we give a method of constructing a relative 𝑍-stable equivalence of Morita type, which is a generalization of stable equivalence of Morita type and was introduced by Wang and Zhang in a more general setting. Then we generalize Linckelmann’s results on stable equivalences of Morita type to relative 𝑍-stable equivalences of Morita type. We also introduce the notion of relative Brauer indecomposability, which is a generalization of the notion of Brauer indecomposability. We give an equivalent condition for Scott modules to be relatively Brauer indecomposable, which is an analog of that given by Ishioka and the first author.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: JP18K03255
Funding statement: This work was supported by JSPS KAKENHI Grant Number JP18K03255.
Acknowledgements
The authors would like to thank the referee for helpful comments.
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Communicated by: Olivier Dudas
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Articles in the same Issue
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- The Higman operations and embeddings of recursive groups
- Redundant relators in cyclic presentations of groups
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- Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
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Articles in the same Issue
- Frontmatter
- The Higman operations and embeddings of recursive groups
- Redundant relators in cyclic presentations of groups
- Commutator endomorphisms of totally projective abelian 𝑝-groups
- Algebraic groups over finite fields: Connections between subgroups and isogenies
- Relative stable equivalences of Morita type for the principal blocks of finite groups and relative Brauer indecomposability
- 5-Regular prime graphs of finite nonsolvable groups
- On weak commutativity in 𝑝-groups
- More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups