List questions from normal-subgroups

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If H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G.

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Munkres' Topology theorem 68.7

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Show the map from the product of disjoint subgroups of G to G is an isomorphism

Published on 03 Jan 2015 - 21:10

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Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$

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What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

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Does N/H=K/H under some terms mean N=K?

Published on 10 Jan 2015 - 14:10

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Operation with normal subgroup

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Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

Published on 13 Jan 2015 - 22:27

492

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True or not: If a normal subgroup and its quotient are commutative, then the group is.

Published on 14 Jan 2015 - 15:12

156

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let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$

Published on 16 Jan 2015 - 22:38

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Check whether two conjugate subgroups are still conjugate in some subgroup

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find about center of G s.t H is normal subgroup of order 2

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Check whether subgroup is normal in $ S_4$

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$x=x^2$ in a sub group?

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List Question

Subgroup Lattice of $G = Z^*_{26}$

If H is a normal subgroup of a group G such that H and G/H are finitely generated, then so is G.

Munkres' Topology theorem 68.7

Show the map from the product of disjoint subgroups of G to G is an isomorphism

Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$

What is the composition series of $\mathbb Z_7$ x $\mathbb Z_{12}$

Does N/H=K/H under some terms mean N=K?

Operation with normal subgroup

Prove there is $\sigma\in S_3$ such that $H_{\sigma (i)} \cong\textrm{}K_i ,\space \forall i $

True or not: If a normal subgroup and its quotient are commutative, then the group is.

let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$

Check whether two conjugate subgroups are still conjugate in some subgroup

find about center of G s.t H is normal subgroup of order 2

Check whether subgroup is normal in $ S_4$

$x=x^2$ in a sub group?

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