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THE NUMBER OF PAIRWISE NONCOMMUTING SETS IN A FINITE GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 17 February 2025, pp. 1-8
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- Article
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We say that two nonempty subsets A and B with cardinality r of a group G are noncommuting subsets if
$xy\neq yx$ for every 
$x\in A$ and 
$y\in B$. We say a nonempty set 
$\mathcal {X}$ of subsets with cardinality r of G is an r-noncommuting set if every two elements of 
$\mathcal {X}$ are noncommuting subsets. If 
$|\mathcal {X}| \geq |\mathcal {Y}|$ for any other r-noncommuting set 
$\mathcal {Y}$ of G, then the cardinality of 
$\mathcal {X}$ (if it exists) is denoted by 
$w_G(r)$ and is called the r-clique number of G. In this paper, we try to find the influence of the function 
$w_G: \mathbb {N} \longrightarrow \mathbb {N}$ on the structure of groups.
