2 results
HALL SUBGROUPS AND
$2$-COCYCLE REGULARITY
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 24 March 2023, pp. 51-56
- Print publication:
- February 2024
-
- Article
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- You have access
- Open access
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Let H be a subgroup of a finite group G and let
$\alpha $ be a complex-valued 
$2$-cocycle of 
$G.$ Conditions are found to ensure there exists a nontrivial element of H that is 
$\alpha $-regular in 
$G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of 
$C_G(x)$ to determine the 
$\alpha $-regularity of a given 
$x\in G.$ In particular, this result implies that every 
$\alpha _H$-regular element of a normal Hall subgroup H is 
$\alpha $-regular in 
$G.$
STRICT REGULARITY FOR
$2$-COCYCLES OF FINITE GROUPS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 24 March 2023, pp. 45-50
- Print publication:
- February 2024
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
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Let
$\alpha $ be a complex-valued 
$2$-cocycle of a finite group 
$G.$ A new concept of strict 
$\alpha $-regularity is introduced and its basic properties are investigated. To illustrate the potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible 
$\alpha _N$-characters of N equals the number of 
$\alpha $-regular conjugacy classes of G contained in 
$N,$ where N is a normal subgroup of 
$G.$
