# Questions tagged [group-isomorphism]

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. In a sense, the existence of such an isomorphism says that the two groups are "the same."

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### Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?

I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt: I first proved that the group $G$, if it exists, cannot be finite ...
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### Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? ("Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki)

I am reading "Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki. Problem 1.17 Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? My attempt: ...
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### Is the direct product of finitely-generated groups cancellative? [duplicate]

The direct product is cancellative for finite groups, so I wanted to know if this result holds for finitely-generated groups as well. The proof linked clearly doesn't apply there, but I have been ...
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### Prove that $S_4/K \cong S_3$ using the fundamental theorem on homomorphism.

Let $A$ be the set formed by the elements of Klein group but identity, A= { (1,2)(3,4);(1,3)(2,4);(1,4)(2,3)}. Consider the set $Big(A)$ of bijection from A to itseld. With the operation of ...
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### Natural map of automorphism groups

Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
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### Use of correspondence theorem for groups to prove that $o(N) = 2$

Let $G$ be a group and $H \triangleleft G$ simple such that $[G : H] = 2$. I have to prove that if $N \neq \{1\}$, $N \triangleleft G$ and $N \cap H = \{1\}$ then $o(N) = 2$. I know by third ...
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### Let $K\lhd G$ be s.t. both $K$ and $G/K$ are simple. Show that either $K$ is the only proper normal subgroup of $G$, or $G \cong K \times (G / K)$.

Sorry about the title, I couldn't fit the whole exercise (Exercise 8.1.6, Nicholson Introduction to Abstract Algebra 4th edition): Let $K \triangleleft G$ be such that both $K$ and $G/K$ are simple. ...
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### Example for $[G : G\cap H] \neq [H:G\cap H]$ with isomorphic subgroups $G \cong H$?

For isomorphic finite subgroups $G,H$ of a group $A$ it holds $[G : G\cap H] = [H: G\cap H]$, since $[G : G\cap H] = \frac{|G|}{|G\cap H|} = \frac{|H|}{|G\cap H|} =[H: G\cap H]$. Does this also hold ...
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### If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$, is then also $G/H \cong G'/H'$?

This seems pretty trivial, but I used it in an assignment and want to double-, triple check. If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$ are normal ...
### "Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show there is $H\le G$ s.t. $G=N\times H$." Does this $=$ really mean $\cong$?
I'm attempting Problem 60 on this problem set. Let $G$ be a finite group with a normal subgroup $N\cong S_3$. Show that there is a subgroup $H$ of $G$ such that $G = N \times H$. I'm guessing the ...
Let $G$ be a group with a normal subgroup $N$ of order $7$ such that $G/N$ is isomorphic to the dihedral group of order $10$. Prove the following: (a) $G$ has a normal subgroup of order $35$. (b) $G$ ...