# 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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### Show there is no isomorphism between a subgroup of $\mathbb{Q}$ and $\mathbb{Z} \times \mathbb{Z}$ [duplicate]

I am currently working on a problem trying to prove that there is no subgroup $H$ of $\mathbb{Q}$ such that $H \cong \mathbb{Z} \times \mathbb{Z}$. I was able to show that $\mathbb{Q}$ cannot be ...
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### $H,K \leq G$, $|H| = 126, |K| = 228$. Show that $H \cap K \leq G$ is either cyclic or isomorphic to $D_3$

So, I don't have to prove that $H \cap K$ is a subgroup of $G$, and $G$ is finite. I know that any group of order $6$ is isomorphic to either $D_3$ or $\mathbb{Z}_6$ (which is cyclic) so I'm assuming ...
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### Character tables of isomorphic groups

Let us consider two groups $G$ and $G'$ which are isomorphic to each other. Since isomorphic groups can be considered as same upto isomorphism, is the character table same for both groups?
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### Let $h\in G$. Let $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$. Then $G$ is abelian iff $\phi_h = \operatorname{id}_G$ for all $h\in G$. [closed]

Let $G$ be a group and $h\in G$. Consider the map $\phi_h : G \to G$ st $g\mapsto hgh^{−1}$, $G$ is an abelian group if and only if $\phi_h = \operatorname{id}_G$ for all $h\in G$. I know what it ...
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I am doing a tutorial question where it ask whether or not the groups of order 4 $(\mathbb{Z}_4,+\mod 4)$ and $(U(5),\times _5)$ are isomorphic to each other. I drew out the Cayley tables and got $$\... 1 vote 2 answers 88 views ### Let G be a group isomorphic to Z_{n_1} \oplus Z_{n_2} \oplus \cdots\oplus Z_{n_k}. Gallian's "Contemporary Abstract Algebra", Chapter 8 Problem 44: Let G be a group isomorphic to Z_{n_1} \oplus Z_{n_2} \oplus \cdots \oplus Z_{n_k}. Let x be the product of all ... 14 votes 1 answer 876 views ### Is this group isomorphic to the real numbers? I have a locally compact abelian group G with the following properties: It is connected (therefore divisible) and non-compact; It admits a \mathbb{Q}-vector space structure and for any g \neq 0,... 1 vote 1 answer 66 views ### Image of a group element under isomorphism I'm not completely sure if this proof works that an isomorphism preserves the order of elements. Here is my attempt at a proof. Let f: G \to H be an isomorphism of groups and x \in G. Then$$\...
Why are all bijective group homomorphisms $\phi: G \to H$ automatically isomorphisms? That is, their inverses are also homomorphisms. I have seen this fact mentioned before, but never a proof. We ...