Shen's conjecture on groups with given same order type

2015/05/31 by L. Jafari Taghvasani, Taghvasani, L. Jafari, M. Zarrin +1
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1506.00199

arxiv created 2015/05/31 · arxiv updated 2015/06/02

Abstract

For any group G, we define an equivalence relation \thicksim as below: ∀ g, h ∈ G g\thicksim h \Longleftrightarrow |g|=|h| the set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denote by α(G). In this paper, we give a partial answer to a conjecture raised by Shen. In fact, we show that if G is a nilpotent group, then |π(G)|≤ |α(G)|, where π(G) is the set of prime divisors of order of G. Also we investigate the groups all of whoseproper subgroups, say H have |α(H)|≤ 2.

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