Question

Prove that if $G$ has no subgroups of index $2$, then every subgroup of index $3$ is normal

score 1k · Answer 1 · 2015-02-02 13:49:28

1k

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Prove that if $G$ has no subgroups of index $2$, then every subgroup of index $3$ is normal

Published on 02 Feb 2015 - 13:49

score 84 · Answer 2 · 2015-02-03 08:52:35

84

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Is there any group in which number of the normal subgroups is equal to number of the conjugacy classes?

Published on 03 Feb 2015 - 8:52

score 50 · Answer 3 · 2015-02-03 19:59:23

50

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If [H,N]≠1 then H∩N≠1 and N≤H . Now NOp′(H) ,so that N∩Op′(H)=1. Hence N is a p-group. Now can we say CH(N)≥Op′(H) and H/CH(N)is p-group?

Published on 03 Feb 2015 - 19:59

score 1.7k · Answer 4 · 2015-02-04 19:54:43

1.7k

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Let $G$ be a group and $H,K,L$ be subgroups with $H \subseteq K$. Prove that $K \cap HL=H(K\cap L)$

Published on 04 Feb 2015 - 19:54

score 387 · Answer 5 · 2015-02-09 19:47:15

387

Views

Prove: $aH = bH \iff Ha^{-1} = Hb^{-1}$

Published on 09 Feb 2015 - 19:47

score 209 · Answer 6 · 2015-02-16 12:51:46

209

Views

The intersection of subgroup and normal subgroup: the greatest common divisor?

Published on 16 Feb 2015 - 12:51

score 1.9k · Answer 7 · 2015-02-16 14:29:01

1.9k

Views

If we have a subgroup of index 3 that is not normal, show there is a subgroup with index 2

Published on 16 Feb 2015 - 14:29

score 139 · Answer 8 · 2015-02-16 18:26:27

139

Views

Classifying groups of order $8$

Published on 16 Feb 2015 - 18:26

score 43 · Answer 9 · 2015-02-16 19:47:53

43

Views

Why is a semigroup $H$ of prime size with proper subgroup $G$ not group?

Published on 16 Feb 2015 - 19:47

score 665 · Answer 10 · 2015-02-17 19:34:00

665

Views

Can a normal subgroup of a finite nonabelian group be nonabelian?

Published on 17 Feb 2015 - 19:34

score 746 · Answer 11 · 2015-02-18 17:59:55

746

Views

Let $K<G$, where G is a group. Define $N_G(K)=(g\in G/gkg^{-1}=K)$ Prove the following:

Published on 18 Feb 2015 - 17:59

score 141 · Answer 12 · 2015-02-19 00:01:02

141

Views

Suppose that $G$ is a group and $A \lhd G$. Prove that if $A$ is abelian and $G=AH$ for some subgroup $H$ of $G$, then $A \cap H \lhd G$

Published on 19 Feb 2015 - 0:01

score 112 · Answer 13 · 2015-02-22 08:10:43

112

Views

Why is it necessary for a quotient group to be defined for a normal subgroup.

Published on 22 Feb 2015 - 8:10

score 770 · Answer 14 · 2026-03-30 23:13:31

770

Views

Prove: $G/(N_1 \cap N_2)$ is isomorphic with a subgroup of $(G/N_1) \times (G/N_2)$

Published on 30 Mar 2026 - 23:13

score 61 · Answer 15 · 2015-02-23 18:04:49

61

Views

Group and subgroup generated by element in G and application of Lagrange' s theorem

Published on 23 Feb 2015 - 18:04

List Question

Prove that if $G$ has no subgroups of index $2$, then every subgroup of index $3$ is normal

Is there any group in which number of the normal subgroups is equal to number of the conjugacy classes?

If [H,N]≠1 then H∩N≠1 and N≤H . Now NOp′(H) ,so that N∩Op′(H)=1. Hence N is a p-group. Now can we say CH(N)≥Op′(H) and H/CH(N)is p-group?

Let $G$ be a group and $H,K,L$ be subgroups with $H \subseteq K$. Prove that $K \cap HL=H(K\cap L)$

Prove: $aH = bH \iff Ha^{-1} = Hb^{-1}$

The intersection of subgroup and normal subgroup: the greatest common divisor?

If we have a subgroup of index 3 that is not normal, show there is a subgroup with index 2

Classifying groups of order $8$

Why is a semigroup $H$ of prime size with proper subgroup $G$ not group?

Can a normal subgroup of a finite nonabelian group be nonabelian?

Let $K<G$, where G is a group. Define $N_G(K)=(g\in G/gkg^{-1}=K)$ Prove the following:

Suppose that $G$ is a group and $A \lhd G$. Prove that if $A$ is abelian and $G=AH$ for some subgroup $H$ of $G$, then $A \cap H \lhd G$

Why is it necessary for a quotient group to be defined for a normal subgroup.

Prove: $G/(N_1 \cap N_2)$ is isomorphic with a subgroup of $(G/N_1) \times (G/N_2)$

Group and subgroup generated by element in G and application of Lagrange' s theorem

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