# 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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### Is there a straightforward group homeomorphism between the 2-adic integers and $[0,1)$?

I think the (set of the) ring of 2-adic integers is represented by: $$\left\{\sum_{i=0}^\infty 2^ix_i:x_i\in\{0,1\}\right\}$$ And I think the set of real numbers in the interval $[0,1)$ is represented ...
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### Number of subgroups of $S_4$ isomorphic to $K_4$

I was trying to find the number of subgroups in $S_4$ which are isomorphic to the Klein's four group $K_4$. I know for doing this, I will have to find the subgroups of the type {$e, a, b, ab$} in $S_4$...
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### The structure of $\textrm{SL}_2(\mathbb{Z}/4\mathbb{Z})$

In Keith Conard's notes https://kconrad.math.uconn.edu/blurbs/grouptheory/SL(2,Z).pdf Page 8, he has stated a fact that $\textrm{SL}_2(\mathbb{Z}/4\mathbb{Z})\cong A_4\rtimes\mathbb{Z}/4\mathbb{Z}$. ...
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### New Criteria of Isomorphism + Abelian Group [duplicate]

The following questions were suggested by my friend, while we were studying fundamental group theory. We had no exact ideas of the way to approach the problems. Questions (1) Let $G$ and $H$ be ...
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### Condition of group isomorphism [duplicate]

If $2$ groups $(G,\times)$ and $(G',*)$ have the same number of subgroups of order $k$, for all positive integers $k$ does it mean that they are isomorphic ?
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### Showing that if $\phi:\Bbb{Z}\oplus\Bbb{Z}\to\Bbb{Z}\oplus\Bbb{Z}$ is an epimorphism of abelian groups, then it is an isomorphism.

I am a mathematician working in analysis and my knowledge in algebra is rusty. Is there a direct argument showing that if $\phi:\mathbb{Z}\oplus\mathbb{Z}\to\mathbb{Z}\oplus\mathbb{Z}$ is an ...
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### Isomorphism between $SO_2\tilde{\times}\mathbb{Z}_2$ and $O_2$

This is the exercise 23.10 p. 135 of Groups and symmetry of Armstrong : Let $G$ be an abelian group and write $G \tilde{\times}\mathbb{Z}_2$ for the semidirect product $G\rtimes_\phi\mathbb{Z}_2$, ...
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### isomorphism between a field and non-field rings [closed]

I think that once you have a field and a ring which is not a field, you can conclude that there is no isomorphism between these two. Is it right? if not, is there an example? if true, can someone ...
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### Does a deformation retraction of $X$ onto a subspace $A\subset X$ induce an isomorphism $\pi_n(X) \to \pi_n(A)$?

Let's say we have a topological space $X$ and a subspace $A\subset X$. Assume $A$ is a deformation retraction of $X$. Does that imply that the induced homomorphism of the deformation retraction is an ...
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### How to prove that some specific group is not isomorphic to any member of any of the 5 families of groups?

For example, let's say we have a group Q8 (Quaternion group). How to prove that this group is not isomorphic to any member of any of the families of groups (Cyclic, Abelian, Dihedral, Symmetric, ...
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### Finding isomorphism between two groups

I'm reading Vern Paulsen's notes. I don't know how can I show $\Bbb R^*/\Bbb R^+$ and $\Bbb Z_2$ are isomorphic because I'm not sure about operation of $\Bbb R^*/\Bbb R^+$. Should I consider it as ...
### Prove that $(\Bbb Z_2 \times \Bbb Z_2\times \Bbb Z_2, +)$ is not isomorphic to $(\Bbb Z_4 \times \Bbb Z_2, +)$
Prove that $(\Bbb Z_2 \times \Bbb Z_2\times \Bbb Z_2, +)$ is not isomorphic to $(\Bbb Z_4 \times \Bbb Z_2, +).$ I believe that both groups have the same cardinality, however, it is not injective as ...