Question

$G\times H$ is not a simple group for $|G|, |H|\geq 2$

score 48 · Answer 1 · 2015-10-28 01:25:02

48

Views

$G\times H$ is not a simple group for $|G|, |H|\geq 2$

Published on 28 Oct 2015 - 1:25

score 377 · Answer 2 · 2026-03-25 10:58:18

377

Views

Groups with all maximal subgroups normal

Published on 25 Mar 2026 - 10:58

score 1k · Answer 3 · 2026-03-18 23:50:50

1k

Views

Show the sub group $\langle r^k\rangle$ is normal in the dihedral group of order $2n$ when $k | n$ and $r$ is the rotation by $2\pi/n$

Published on 18 Mar 2026 - 23:50

score 142 · Answer 4 · 2015-10-29 21:07:09

142

Views

Proving $H\triangleleft G$ for a finite group $G$ and a subgroup $H$

Published on 29 Oct 2015 - 21:07

score 2.5k · Answer 5 · 2026-04-01 00:26:53

2.5k

Views

If $N$ and $M$ are normal subgroups of $G$, show that $NM/M$ is isomorphic to $N/N \cap M$.

Published on 01 Apr 2026 - 0:26

score 225 · Answer 6 · 2015-11-02 09:44:58

225

Views

Commutator subgroup is the minimal normal subgroup such that quotient group is abelian

Published on 02 Nov 2015 - 9:44

score 85 · Answer 7 · 2015-11-02 17:07:53

85

Views

If $K$ is a normal subgroup of $G$, is it true that $G$ is isomorphic to the direct product $K \times (G / K)$?

Published on 02 Nov 2015 - 17:07

score 3.7k · Answer 8 · 2026-04-01 00:28:28

3.7k

Views

The preimage of a normal subgroup under a group homomorphism is normal

Published on 01 Apr 2026 - 0:28

score 14.7k · Answer 9 · 2015-11-03 17:05:37

Prove that if H is a normal subgroup of G and K is a normal subgroup of H, then K may not be a normal subgroup of G.

score 60 · Answer 10 · 2026-04-01 00:51:18

60

Views

Why is $\Bbb{Z_2} / \{e\} = \Bbb{Z_2}$?

Published on 01 Apr 2026 - 0:51

score 977 · Answer 11 · 2026-04-01 00:55:21

977

Views

Finite group is solvable iff it has a normal series with $p$-primary abelian quotient groups

Published on 01 Apr 2026 - 0:55

score 184 · Answer 12 · 2015-11-08 06:03:05

184

Views

Normal Subgroup and Isomorphism of a Quotient group

Published on 08 Nov 2015 - 6:03

score 710 · Answer 13 · 2026-04-01 13:39:24

710

Views

If $H \triangleleft G$, and $H$ is maximal, then $G/H$ is simple

Published on 01 Apr 2026 - 13:39

score 53 · Answer 14 · 2015-11-09 13:45:36

53

Views

If $H\unlhd G$ and $ K\unlhd G$ and $ H \cap K= \{e\}$. Prove $xy=yx \forall x \in H \forall y \in K$

Published on 09 Nov 2015 - 13:45

score 121 · Answer 15 · 2015-11-11 23:06:35

121

Views

Understanding how a subgroup is normal if and only if the relation $\equiv$ (N) respects products and inverses

Published on 11 Nov 2015 - 23:06

List Question

$G\times H$ is not a simple group for $|G|, |H|\geq 2$

Groups with all maximal subgroups normal

Show the sub group $\langle r^k\rangle$ is normal in the dihedral group of order $2n$ when $k | n$ and $r$ is the rotation by $2\pi/n$

Proving $H\triangleleft G$ for a finite group $G$ and a subgroup $H$

If $N$ and $M$ are normal subgroups of $G$, show that $NM/M$ is isomorphic to $N/N \cap M$.

Commutator subgroup is the minimal normal subgroup such that quotient group is abelian

If $K$ is a normal subgroup of $G$, is it true that $G$ is isomorphic to the direct product $K \times (G / K)$?

The preimage of a normal subgroup under a group homomorphism is normal

Prove that if H is a normal subgroup of G and K is a normal subgroup of H, then K may not be a normal subgroup of G.

Why is $\Bbb{Z_2} / \{e\} = \Bbb{Z_2}$?

Finite group is solvable iff it has a normal series with $p$-primary abelian quotient groups

Normal Subgroup and Isomorphism of a Quotient group

If $H \triangleleft G$, and $H$ is maximal, then $G/H$ is simple

If $H\unlhd G$ and $ K\unlhd G$ and $ H \cap K= \{e\}$. Prove $xy=yx \forall x \in H \forall y \in K$

Understanding how a subgroup is normal if and only if the relation $\equiv$ (N) respects products and inverses

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