A family of finite p‐groups satisfying Carlson's depth conjecture. Issue 6 (24th April 2022)

• Record Type: Journal Article
• Authors: Garaialde Ocaña, Oihana; González‐Sánchez, Jon; Guerrero Sánchez, Lander
• Is Part Of: Mathematische Nachrichten. Volume 295:Issue 6(2022)
• Page Start: 1174
• Page End: 1185
• Publication Date: 2022-04-24
• Subjects: depth -- finite p‐groups -- mod‐p cohomology ring; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.202000551
• Languages: English
• ISSNs: 0025-584X

Abstract

Abstract: Let p > 3 $p>3$ be a prime number and let r be an integer with 1 < r < p − 1 $1<r<p-1$ . For each r, let moreover G r $G_r$ denote the unique quotient of the maximal class pro‐ p group of size p r + 1 $p^{r+1}$ . We show that the mod‐ p cohomology ring of G r $G_r$ has depth one and that, in turn, it satisfies the equalities in Carlson's depth conjecture [2]. This is the first family of finite p ‐groups for which Carlson's depth conjecture has been verified besides p ‐groups of abelian type mod‐ p cohomology or extraspecial p ‐groups. Moreover, this computation is possible without first describing the structure of the cohomology ring.