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| Paper IPM / M / 8727 |
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| Abstract: | |
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The commuting graph of a ring R, denoted by Γ(R), is a
graph whose vertices are all non-central elements of R and two
distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(R),
is similarly defined. In this paper we investigate some
graph-theoretic properties of Γ(Mn(F)), where F is a
field and n\geqslant 2. Also we study the commuting graphs of
some classical groups such as GLn(F) and SLn(F). We show
that Γ(Mn(F)) is a connected graph if and only if every
field extension of F of degree n contains a proper
intermediate field. We prove that apart from finitely many fields,
a similar result is true for Γ(GLn(F)) and
Γ(SLn(F)). Also we show that for two fields F and E
and integers n,m \geqslant 2, if Γ(Mn(F)) ≅ Γ(Mm(E)), then n = m and |F| = |E|.
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