{"paper":{"title":"Power maps in finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.NT"],"primary_cat":"math.CO","authors_text":"Matt Larson","submitted_at":"2017-07-20T20:08:27Z","abstract_excerpt":"In recent work, Pomerance and Shparlinski have obtained results on the number of cycles in the functional graph of the map $x \\mapsto x^a$ in $\\mathbb{F}_p^*$. We prove similar results for other families of finite groups. In particular, we obtain estimates for the number of cycles for cyclic groups, symmetric groups, dihedral groups and $SL_2(\\mathbb{F}_q)$. We also show that the cyclic group of order $n$ minimizes the number of cycles among all nilpotent groups of order $n$ for a fixed exponent. Finally, we pose several problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06696","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}