Question

Is it true that $aH = bH$ iff $ab^{-1} \in H$

score 3.7k · Answer 1 · 2015-04-01 17:23:19

3.7k

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Is it true that $aH = bH$ iff $ab^{-1} \in H$

Published on 01 Apr 2015 - 17:23

score 1.1k · Answer 2 · 2026-03-26 14:31:58

1.1k

Views

Prove that if H and K are normal nilpotent subgroups of a finite group G, then HK is a normal nilpotent subgroup.

Published on 26 Mar 2026 - 14:31

score 763 · Answer 3 · 2015-04-02 11:32:16

763

Views

Find the order of $Z(G)$.

Published on 02 Apr 2015 - 11:32

score 981 · Answer 4 · 2015-04-05 08:46:44

981

Views

Prove that $H$ is a normal subgroup of $G$. [$aH =Hb$]

Published on 05 Apr 2015 - 8:46

score 96 · Answer 5 · 2015-04-08 12:31:55

96

Views

If $K$ and $H$ are subgroups of $G$ and $H \triangleleft K$ then $K \subseteq N(H)$.

Published on 08 Apr 2015 - 12:31

score 752 · Answer 6 · 2026-03-30 02:08:54

752

Views

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Published on 30 Mar 2026 - 2:08

score 919 · Answer 7 · 2015-04-10 21:23:57

919

Views

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$.

Published on 10 Apr 2015 - 21:23

score 27 · Answer 8 · 2026-03-31 05:41:34

27

Views

Consider $\phi: G \to S_4\times D_{15}$ a homomorphism and onto

Published on 31 Mar 2026 - 5:41

score 10.9k · Answer 9 · 2015-04-14 01:07:59

10.9k

Views

When is a group isomorphic to a proper subgroup of itself?

Published on 14 Apr 2015 - 1:07

score 3.9k · Answer 10 · 2015-04-16 19:56:39

3.9k

Views

Prove that the union of two normal subgroups is also a normal subgroup

Published on 16 Apr 2015 - 19:56

score 229 · Answer 11 · 2026-03-29 23:58:46

229

Views

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Published on 29 Mar 2026 - 23:58

score 2.1k · Answer 12 · 2015-04-19 18:47:41

2.1k

Views

Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

Published on 19 Apr 2015 - 18:47

score 606 · Answer 13 · 2015-04-20 07:34:14

606

Views

Case when all subgroups of a group are normal

Published on 20 Apr 2015 - 7:34

score 166 · Answer 14 · 2015-04-20 14:45:07

166

Views

If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

Published on 20 Apr 2015 - 14:45

score 684 · Answer 15 · 2015-04-20 15:10:22

684

Views

Is $A_4$ isomorphic to $D_3\times \mathbb Z_2$?

Published on 20 Apr 2015 - 15:10

List Question

Is it true that $aH = bH$ iff $ab^{-1} \in H$

Prove that if H and K are normal nilpotent subgroups of a finite group G, then HK is a normal nilpotent subgroup.

Find the order of $Z(G)$.

Prove that $H$ is a normal subgroup of $G$. [$aH =Hb$]

If $K$ and $H$ are subgroups of $G$ and $H \triangleleft K$ then $K \subseteq N(H)$.

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$.

Consider $\phi: G \to S_4\times D_{15}$ a homomorphism and onto

When is a group isomorphic to a proper subgroup of itself?

Prove that the union of two normal subgroups is also a normal subgroup

How many elements does the free product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/2\mathbb{Z}$ have?

Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

Case when all subgroups of a group are normal

If $G$ is a finite group such that the Klein four group is a quotient of $G$ , then can we write $G$ as a union of three proper subgroups ?

Is $A_4$ isomorphic to $D_3\times \mathbb Z_2$?

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