Question

Let $G=\langle a,b\rangle$. Prove neither $\langle a\rangle$ nor $\langle b \rangle$ are normal in $G,$ for $a=(1234)(57), b=(24)(5678)$.

score 74 · Answer 1 · 2026-03-26 12:37:49

74

Views

Let $G=\langle a,b\rangle$. Prove neither $\langle a\rangle$ nor $\langle b \rangle$ are normal in $G,$ for $a=(1234)(57), b=(24)(5678)$.

Published on 26 Mar 2026 - 12:37

score 622 · Answer 2 · 2020-05-09 12:45:37

622

Views

Is there a simple formula for the number of subgroups of index 2 of $\mathbb{Z}_2^n$?

Published on 09 May 2020 - 12:45

score 858 · Answer 3 · 2026-03-26 14:22:56

858

Views

If $H_1$ and $H_2$ are isomorphic normal subgroups of $G$, when do we have an isomorphism between $G/H_1$ and $G/H_2$?

Published on 26 Mar 2026 - 14:22

score 50 · Answer 4 · 2020-05-11 04:09:23

50

Views

Is$ \langle G,+\rangle$ a sub-group of $\langle F,+\rangle$?

Published on 11 May 2020 - 4:09

score 338 · Answer 5 · 2026-03-26 02:57:42

338

Views

$G$ is a group with a normal subgroup $K$ such that $G/K$ is soluble, and $H$ is a nonabelian simple subgroup of $G$, then $H \leq K$

Published on 26 Mar 2026 - 2:57

score 165 · Answer 6 · 2020-05-12 00:03:43

165

Views

the Frattini subgroup of the Fitting subgroup of a group whose Frattini subgroup is trivial

Published on 12 May 2020 - 0:03

score 147 · Answer 7 · 2026-03-27 11:47:59

147

Views

Why is $C_G(A)$ a normal subgroup of $B$ in this context?

Published on 27 Mar 2026 - 11:47

score 54 · Answer 8 · 2026-03-25 21:04:04

54

Views

$[G,G] \leq \langle g_1,...,g_{n-1}\rangle$ if and only if $\langle g_1,...,g_{n-1}\rangle$ is normal in $G$

Published on 25 Mar 2026 - 21:04

score 324 · Answer 9 · 2020-05-13 00:59:13

324

Views

Showing that any two elements in two normal subgroups commute

Published on 13 May 2020 - 0:59

score 52 · Answer 10 · 2026-03-26 17:46:02

52

Views

Using Definition of Cyclic Group to prove B is a Subgroup

Published on 26 Mar 2026 - 17:46

score 97 · Answer 11 · 2020-05-14 14:34:48

97

Views

Is a subgroup test necessary if proving normality?

Published on 14 May 2020 - 14:34

score 115 · Answer 12 · 2020-05-15 09:15:32

115

Views

If $K \trianglelefteq G$ has prime index $p$, show that $\exists a \in G$ such that $G=$ $K \cup K a \cup \cdots \cup K a^{p-1}$.

Published on 15 May 2020 - 9:15

score 318 · Answer 13 · 2020-05-17 13:27:24

318

Views

On the proof of classification of finitely generated abelian groups

Published on 17 May 2020 - 13:27

score 289 · Answer 14 · 2026-03-25 09:22:51

289

Views

Lie Group structure on $S^2$

Published on 25 Mar 2026 - 9:22

score 71 · Answer 15 · 2020-05-18 10:20:39

71

Views

Given a subgroup $B$ of a group $A$, show that $a\in B$ iff $Ba = B$

Published on 18 May 2020 - 10:20

List Question

Let $G=\langle a,b\rangle$. Prove neither $\langle a\rangle$ nor $\langle b \rangle$ are normal in $G,$ for $a=(1234)(57), b=(24)(5678)$.

Is there a simple formula for the number of subgroups of index 2 of $\mathbb{Z}_2^n$?

If $H_1$ and $H_2$ are isomorphic normal subgroups of $G$, when do we have an isomorphism between $G/H_1$ and $G/H_2$?

Is$ \langle G,+\rangle$ a sub-group of $\langle F,+\rangle$?

$G$ is a group with a normal subgroup $K$ such that $G/K$ is soluble, and $H$ is a nonabelian simple subgroup of $G$, then $H \leq K$

the Frattini subgroup of the Fitting subgroup of a group whose Frattini subgroup is trivial

Why is $C_G(A)$ a normal subgroup of $B$ in this context?

$[G,G] \leq \langle g_1,...,g_{n-1}\rangle$ if and only if $\langle g_1,...,g_{n-1}\rangle$ is normal in $G$

Showing that any two elements in two normal subgroups commute

Using Definition of Cyclic Group to prove B is a Subgroup

Is a subgroup test necessary if proving normality?

If $K \trianglelefteq G$ has prime index $p$, show that $\exists a \in G$ such that $G=$ $K \cup K a \cup \cdots \cup K a^{p-1}$.

On the proof of classification of finitely generated abelian groups

Lie Group structure on $S^2$

Given a subgroup $B$ of a group $A$, show that $a\in B$ iff $Ba = B$

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