On the size of the Schur multiplier of finite groups
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| Format: | Preprint |
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2023
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| author | Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. |
| author_facet | Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. |
| contents | We obtain bounds for the size of the Schur multiplier of finite $p$-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group $H^2(G,\mathbb{Z}/p\mathbb{Z})$ of a $p$-group with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Denoting the minimal number of generators of a $p$-group $G$ by $d(G)$, our bound depends on the parameters $|G|=p^n$, $|γ_2G|=p^k$, $d(G)=d$, $d(G/Z)=δ$ and $d(γ_2G/γ_3G)=k'$. For special $p$-groups, we further improve our bound when $δ-1 > k'$. Moreover, given natural numbers $d$, $δ$, $k$ and $k'$ satisfying $k=k'$ and $δ-1 \leq k'$, we construct a capable $p$-group $H$ of nilpotency class two and exponent $p$ such that the size of the Schur multiplier attains our bound. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_02793 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the size of the Schur multiplier of finite groups Kalithasan, Sathasivam Mavely, Tony N. Thomas, Viji Z. Group Theory Combinatorics 15A63, 15A69, 11E39, 20D15, 20F18, 20J05, 20J06 We obtain bounds for the size of the Schur multiplier of finite $p$-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group $H^2(G,\mathbb{Z}/p\mathbb{Z})$ of a $p$-group with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Denoting the minimal number of generators of a $p$-group $G$ by $d(G)$, our bound depends on the parameters $|G|=p^n$, $|γ_2G|=p^k$, $d(G)=d$, $d(G/Z)=δ$ and $d(γ_2G/γ_3G)=k'$. For special $p$-groups, we further improve our bound when $δ-1 > k'$. Moreover, given natural numbers $d$, $δ$, $k$ and $k'$ satisfying $k=k'$ and $δ-1 \leq k'$, we construct a capable $p$-group $H$ of nilpotency class two and exponent $p$ such that the size of the Schur multiplier attains our bound. |
| title | On the size of the Schur multiplier of finite groups |
| topic | Group Theory Combinatorics 15A63, 15A69, 11E39, 20D15, 20F18, 20J05, 20J06 |
| url | https://arxiv.org/abs/2309.02793 |