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On Camina Pairs in Dicyclic Groups and the Heisenberg Group | ||
| International Journal of Group Theory | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 26 مهر 1405 | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2026.149447.2030 | ||
| نویسندگان | ||
| Maryam Mehrdad1؛ Mohamad reza Darafsheh2؛ Hesam Sharifi* 3 | ||
| 1Department of Mathematics and Computer Science, Shahed University, Tehran, Iran | ||
| 2School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran | ||
| 3Shahed University, Tehran, Iran. | ||
| چکیده | ||
| Let $G$ be a finite group and $N$ a proper non-trivial normal subgroup of $G$. The pair $(G,N)$ is called a Camina pair if every $x\in G\setminus N$ is conjugate to all elements of the coset $xN$; when $N=Z(G)$ one speaks of a central Camina pair. In this paper, we classify all Camina pairs arising from dicyclic groups $T_{4n}$. We prove that $(T_{4n},Z(T_{4n}))$ is a central Camina pair if and only if $T_{4n}\cong Q_8$, the quaternion group of order $8$. More generally, we show that a proper non-trivial normal subgroup $N\triangleleft T_{4n}$ forms a Camina pair $(T_{4n},N)$ precisely when $T_{4n}\cong Q_8$ and $N=Z(Q_8)$. The proof combines structural lemmas on conjugacy classes of dicyclic groups with the classical Camina dichotomy. We then turn to the Heisenberg group $UT(3,q)$, the Sylow $p$-subgroup of $GL_{3}(q)$, and confirm the conjecture $[G:Z(G)]\ge|Z(G)|^{2}$ for this family, showing that equality holds. As a consequence, we determine that every non-central conjugacy class in $UT(3,q)$ has size $q$. Finally, we relate central Camina pairs to GVZ-groups via the character-theoretic vanishing condition of Lemma~\ref{lem1}(d), and discuss how this connection suggests further lines of investigation. | ||
| کلیدواژه‌ها | ||
| Camina pairs؛ dicyclic groups؛ Heisenberg group؛ GVZ-groups | ||
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