An antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the integers 1,. . .,m such that all n vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called antimagic if it has an antimagic labeling. A conjecture of Ringel (see [4]) states that every connected graph, but K 2 , is antimagic. Our main result validates this conjecture for graphs having minimum degree (log n). The proof combines probabilistic arguments with simple tools from analytic number theory and combinatorial techniques. We also prove that complete partite graphs (but K 2 ) and graphs with maximum degree at least n À 2 are antimagic.

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We study products of Sylow subgroups of a finite group G. First we prove that G is solvable if and only if G = P 1 · · · P m for any choice of Sylow p i -subgroups P i , where p 1 , . . . , p m are all of the distinct prime divisors of |G|, and for any ordering of the p i . Then, for a general finite group G, we show that the intersection of all Sylow products as above is a subgroup of G which is closely related to the solvable radical of G.

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We call a subgroup A of a finite group G a solitary subgroup of G if G does not contain another isomorphic copy of A. We call a normal subgroup A of a finite group G a normal solitary subgroup of G if G does not contain another normal isomorphic copy of A. The property of being (normal) solitary can be viewed as a strengthening of the requirement that A is normal in G. We derive various results on the existence of (normal) solitary subgroups.