Bounds for the exponent of the Schur multiplier

The exponent of the non‐abelian tensor square and related constructions ofp‐groups

“…The previous result improves the bounds obtained [1, 18, 24] (cf. [1, Theorem 6.5], [18, Section 2] and [24, Theorem 1.1]).…”

“…In [18], Moravec showed that if G is a p ‐group of class $$c\ge 2$$, then $$\exp (M(G))$$ divides $$\exp (G)^{2\lfloor \log _2(c)\rfloor }$$. Later, in [24], Sambonet proved that if G is a p ‐group of class $$c\ge 2$$, then $$\exp (M(G))$$ divides $$\exp (G)^{\big\lfloor \log _{p-1}(c)\big\rfloor +1}$$ if $$p>2$$ and $$\exp (M(G))$$ divides $$2^{\lfloor \log _2(c)\rfloor } \cdot \exp (G)^{\lfloor \log _2(c)\rfloor +1}$$ if $$p=2$$. In [1], Antony et al.…”

“…[20, Corollary 4.8]). Furthermore, in the context of the Sambonet theorem [24, Theorem 3.3], the improvement occurs for $$e\le r$$ if $$p>2$$ and $$e\le r+2$$ if $$p=2$$, where $$\exp (G)=p^e$$.…”

On the exponent of the Weak commutativity group $χ(G)$

“…Set m = ⌊log 2 (c)⌋ and d the derived length of G. According to Theorem 1.1 ii) and the formula d ≤ ⌊log 2 (c)⌋ + 1 (cf. [9, Theorem 5.1.12]), we deduce that exp(D(G)) divides 2 d−1 •exp(G) d and so, exp(D(G)) divides 2 m • exp(G) m+1 , which completes the proof.As an immediate consequence of the above result we obtain Sambonet's theorem[13, Theorem 1.1] concerning the exponent of the Schur multiplier of p-groups of class c. More precisely, we conclude the following extension.…”

Bogomolov Multipliers of P-Groups of Maximal Class

“…More precisely, it is conjectured that exp M(G) divides exp G for a finite p-group p > 2. Confer [25] and the references therein. We show how there is at least a linear bound on the exponent of the Schur multiplier provided that the coclass is fixed.…”