Question

Nilpotent Group and Normal Series

score 112 · Answer 1 · 2021-09-02 13:34:28

112

Views

Nilpotent Group and Normal Series

Published on 02 Sep 2021 - 13:34

score 198 · Answer 2 · 2021-09-02 17:02:24

198

Views

Minimal normal subgroup is contained in Centre of group for nilpotent groups

Published on 02 Sep 2021 - 17:02

score 142 · Answer 3 · 2021-09-04 01:37:00

142

Views

How to use the upper central series instead of the lower one in nilpotency proofs

Published on 04 Sep 2021 - 1:37

score 137 · Answer 4 · 2021-09-07 22:02:54

137

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Constructing central series

Published on 07 Sep 2021 - 22:02

score 102 · Answer 5 · 2021-09-12 19:02:55

102

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Let $G$ be a nilpotent group and $|G| = p_1p_2p_3$ where $p_i$ are different prime numbers. Prove that $G$ is an abelian group.

Published on 12 Sep 2021 - 19:02

score 302 · Answer 6 · 2021-09-12 19:41:09

302

Views

Suppose $T(B)=AB+BA$. Prove that if $A$ is a nilpotent matrix, then $T$ is a nilpotent operator. (Question on Existing Proof)

Published on 12 Sep 2021 - 19:41

score 230 · Answer 7 · 2021-09-12 22:16:26

230

Views

Lower central series of the Unitriangular group $UT(n, \mathbb{Z}_p)$

Published on 12 Sep 2021 - 22:16

score 107 · Answer 8 · 2026-02-23 07:41:59

107

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why "this means that $G'' \subset Z(G')$"?

Published on 23 Feb 2026 - 7:41

score 139 · Answer 9 · 2021-10-22 13:40:08

139

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If all proper subgroups of $G$ are nilpotent and $G$ is solvable, the $G$ is nilpotent.

Published on 22 Oct 2021 - 13:40

score 35 · Answer 10 · 2026-02-23 07:44:15

35

Views

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic?

Published on 23 Feb 2026 - 7:44

score 393 · Answer 11 · 2021-11-29 19:37:52

393

Views

If a nilpotent group has an element of prime order $p$, so does its centre.

Published on 29 Nov 2021 - 19:37

score 97 · Answer 12 · 2021-12-05 14:05:19

97

Views

Show $G=\langle\delta\rangle\ltimes D$ is nilpotent of class $2$.

Published on 05 Dec 2021 - 14:05

score 58 · Answer 13 · 2021-12-06 00:16:04

58

Views

Characterization of nilpotency with normal subgroups

Published on 06 Dec 2021 - 0:16

score 290 · Answer 14 · 2022-01-05 23:36:13

290

Views

If $N$ is a normal subgroup of $G$, show $Z(G)N/N \subset Z(G/N)$

Published on 05 Jan 2022 - 23:36

score 57 · Answer 15 · 2022-01-06 17:28:21

57

Views

If $N$ is normal in $G$, show $Z_{i}(G)N/N \leq Z_{i}(G/N)$ where $Z_{i}(G)$ is the $i$th term in the upper central series for $G$.

Published on 06 Jan 2022 - 17:28

List Question

Nilpotent Group and Normal Series

Minimal normal subgroup is contained in Centre of group for nilpotent groups

How to use the upper central series instead of the lower one in nilpotency proofs

Constructing central series

Let $G$ be a nilpotent group and $|G| = p_1p_2p_3$ where $p_i$ are different prime numbers. Prove that $G$ is an abelian group.

Suppose $T(B)=AB+BA$. Prove that if $A$ is a nilpotent matrix, then $T$ is a nilpotent operator. (Question on Existing Proof)

Lower central series of the Unitriangular group $UT(n, \mathbb{Z}_p)$

why "this means that $G'' \subset Z(G')$"?

If all proper subgroups of $G$ are nilpotent and $G$ is solvable, the $G$ is nilpotent.

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic?

If a nilpotent group has an element of prime order $p$, so does its centre.

Show $G=\langle\delta\rangle\ltimes D$ is nilpotent of class $2$.

Characterization of nilpotency with normal subgroups

If $N$ is a normal subgroup of $G$, show $Z(G)N/N \subset Z(G/N)$

If $N$ is normal in $G$, show $Z_{i}(G)N/N \leq Z_{i}(G/N)$ where $Z_{i}(G)$ is the $i$th term in the upper central series for $G$.

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