# Tagged Questions

Questions concerning normal-subgroups of groups.

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### How can I find all the numbers of order 10 in $Z_{60}$? [on hold]

How can I find all the numbers of order $10$ in $Z_{60}$? In fact, these are all the numbers with $(x,60)=6$. How can I find all these numbers?
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### It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold]

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1
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### Normal subgroup. [duplicate]

Let $N$ be subgroup of a group $G$. Suppose that, for each $a\in G$, there exists $b\in G$ such that $Na=bN$. Prove that $N$ is a normal subgroup. Please guide me with a proof. Thank you for your ...
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### To prove: The intersection of all normal subgroups of finite index of a free group is trivial.

Let $S$ be a set, $G$ its free group and $\mathcal{N}$ the set of normal $N \leq G$ such that $[G:N] \leq \infty$. Prove that $$\bigcap_{N \in \mathcal{N}} N \ = \ \{ e\}$$ I know how free ...
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### Index of intersection of two normal subgroups

Question: Let $G$ be a finite group and $H$ and $K$ are normal subgroup of $G$. If $[G:H]=2$ and $[G:K]=3$, determine $[G:H∩K]$. My attempt: I think $[G:H∩K] = 6$. Since $G$, $H$, $K$ are ...
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### If a subgroup has smallest prime index, then it is normal

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
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### Finding groups $H$ for which there exists surjective homomorphisms $f:D_4 \rightarrow H$?

How can I find out for which groups $H$ there exists surjective homomorphisms $f: D_4 \rightarrow H$? $D_4$ is the dihedral group of the square. I have a theorem that says that there exists such ...
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### Group of order $54$ has normal sugroup of order $27.$

Let $G$ be a group of order $54$. Prove that there exists a normal subgroup of order $27.$ Is this normal subgroup unique? Thoughts. Since $27$ divides $54$, by Lagrange's theorem we can not exclude ...
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### If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$

I wish to prove whether this is true or false. If $\phi: G \rightarrow H$ is a group homomorphism, $N \vartriangleleft G$, then $G/N \cong \phi(G)/\phi(N)$. I'm not even sure if $N$ being normal ...
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### We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that ...
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### Let $G=\langle \mathbb{Z},+\rangle$ and $H=\{6n|n \in \mathbb{Z}\}$. Find all the distinct left and right cosets of $H$ in $G$.

I have an exercise where I am supposed to find the left and right cosets. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set $H$ and ...
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### Determing whether a subgroup is normal

I have been working with normal subgroups and feel like I am doing something wrong. I understand there are many ways to demonstrate if a subgroup is normal, but the methods seem to take longer than I ...
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### Normal subgroups of $A_5$ must contain a 3-cycle.

I am trying to prove the simplicity of $A_5$ by showing that every non-trivial normal subgroup $H$ contains a 3-cycle, and therefore is all of $A_5$ since the 3-cycles all belong to one conjugacy ...
### show that for two normal subgroups with trivial intersection $n_1n_2 = n_2n_1$ [closed]
Let $G$ be a group, and $N_1, N_2$ normal subgroups of $G$ s.t. $N_1\cap N_2 = \{e\}$. Show that $\forall n_1 \in N_1, \forall n_2 \in N_2, n_1n_2 = n_2n_1$.