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null (Ed.)Abstract: Morgan and Parker proved that if G is a group with Z(G)=1, then the connected components of the commuting graph of G have diameter at most 10. Parker proved that if, in addition, G is solvable, then the commuting graph of G is disconnected if and only if G is a Frobenius group or a 2-Frobenius group, and if the commuting graph of G is connected, then its diameter is at most 8. We prove that the hypothesis Z (G) = 1 in these results can be replaced with G' \cap Z(G)=1. We also prove that if G is solvable and G/Z(G) is either a Frobenius group or a 2-Frobenius group, then the commuting graph of G is disconnected.more » « less
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In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$ . In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $$R$$ is an algebra over a field of positive characteristic and $$D$$ is a locally nilpotent derivation.more » « less
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null (Ed.)Abstract: For a group G, we define a graph Delta (G) by letting G^#=G\{1} be the set of vertices and by drawing an edge between distinct elements x,y in G^# if and only if the subgroup is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate Delta (G) for a Z-group G.more » « less
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null (Ed.)We characterize bijective linear maps on [Formula: see text] that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar linear preserver problems.more » « less
