{"paper":{"title":"Fixed points of automorphisms of certain non-cyclic $p$-groups and the dihedral group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Akhtar Abbas, Daniel L\\'opez-Aguayo, Umar Hayat","submitted_at":"2018-01-10T03:36:40Z","abstract_excerpt":"Let $G=\\mathbf{Z}_{p} \\oplus \\mathbf{Z}_{p^2}$, where $p$ is a prime number. Suppose that $d$ is a divisor of the order of $G$. In this paper we find the number of automorphisms of $G$ fixing $d$ elements of $G$, and denote it by $\\theta(G,d)$. As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group $\\mathbf{Z}_{p^{a}} \\oplus \\mathbf{Z}_{p^{b}}$, where $a$ and $b$ are positive integers with $a<b$. Finally, we compute $\\theta(D_{2q},d)$, where $D_{2q}$ is the dihedral group of order $2q$, $q$ is an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.03229","kind":"arxiv","version":7},"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"}