# 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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### Order of automorphisms of non-abelian group

I'm trying to solve the problem, but it doesn't work. Please help me to solve it. Problem: prove that $\bigl| \operatorname{Aut} \, (G) \bigr| \ge 8$ if $G$ is non-abelian and not isomorphic with $S_3$...
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### Show that the intersection of these two subgroups has prime index

Let the subgroup $N$ be normal in the finite group $G$ with index a prime $p$. Let $H$ be a subgroup of $G$ which is not contained in $N$. I would like to show that $H \cap N$ is normal in $H$ with ...
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### Prove that if $H_1\lhd G, H_2\lhd G, H_1H_2=G, H_1\cap H_2=\{1\}$, then $G/H_1\simeq H_2$.

Problem: Suppose $G$ is a group, and $H_1, H_2$ are subgroups of $G$. Prove that if $H_1, H_2$ are normal subgroups of $G$, $H_1H_2=\{xy: x\in H_1, y\in H_2\}=G$, and $H_1\cap H_2=\{1\}$ where $1$ is ...
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### What is the example of non canonical isom? [closed]

In mathematics, isomorphism $f$ between two objects are sometimes called canonical when $f$ is unique as a map, in other words, everyone choose the same map, no different map gives the isomorphism. ...
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### Isomorphisms preserve order: the infinite case

Given a group $G$ and a isomorphism $\varphi: G \to H$, I know how to prove that $|x| = |\varphi(x)|$ in the finite case. If $|x| = n$, I prove that $\varphi(x^n) = \varphi(x)^n$, and since $x^n = e$, ...
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### Find specific example for where the case works (topic: isomorphism and normal subgroups)

Consider groups $G_i, K_i, F_i$ such that $K_i \triangleleft G_i$ and $G_i / K_i\cong F_i$ , for $i = 1, 2$. In each case, find an example with the given properties or prove that no such example ...
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### question for group theory [duplicate]

hello i am korean student of mathematical education. first of all, i am sorry for lack of my english skills. here is question. suppose H, K is normal subgroup of group G. If H is isomorphic to K, is ...
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### Classifying groups of order less than or equal to $4$ up to isomorphism

I'm trying to prove the following assertions. Classify groups of order less than or equal to $4$ along the following lines: (a) There is one group of order $2$ up to isomorphism. (b) There is one ...
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### Question about isomorphism between the subgroups $N$ and $a^{-1}Na$

In showing isomorphism between the subgroups $N$ and $a^{-1}Na$ of a group $G$, one usually define a function $f:N\rightarrow a^{-1}Na$ by $f(n)=a^{-1}na$ for all $a\in G$ and some $n\in N$. This ...
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### Why does injectivity imply to $|G/(H \cap K)|\leq |G/H|\cdot|G/K|?$

I'm looking through some proof about the inequality in the title, the one defines: $$\phi: G/(H \cap K)\rightarrow G/H\times G/K$$ $$\phi(g(H \cap K))=(gH,gK)$$ Note that $\phi$ is injective, I'd like ...
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### Finding isomorphisms among groups

The lecture notes I'm working through pose an exercise to find groups that are isomorphic to each other or to subgroups of other groups. The groups listed so far are: The trivial group $\{e\}$ ...
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### Quotient group by normal closure of union

Let $G$ be a group and $H, N \subseteq G$ be subsets of the group. Let $\overline{A}$ denote the normal closure of any subset $A\subseteq G'$ in some group $G'$. Let $\pi: G \to G/\overline{N}$ denote ...
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### The given function $\phi$ is not an isomorphism because it is not surjective. But it's not injective either, right?

Here's the problem I did for homework from A First Course in Abstract Algebra, 7th Edition by John B. Fraleigh. I just want to check if my reasoning is correct on problem number 15 from Section 3: ...
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### What is wrong with my proof of group order?

Let $\phi: G\rightarrow H$ be an isomorphism of groups and let $a \in G$ be of order $n$. Show that the order of $\phi(a)$ is also $n$. I was given this problem a week ago during a quiz and my ...
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### If $G,H$ are finite groups, then $G\times G\cong H \times H$ implies $G \cong H$

Proposition. Let $G,H$ be finite groups (abelian or not). Then the following implication holds: $$G\times G\cong H\times H \Rightarrow G\cong H.$$ In the case of $G,H$ both abelian, one can use the ...
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### Prove that ${\rm Inn}(S_n)$ isomorphic to $S_n.$

Show that ${\rm Inn}(S_n)$ isomorphic to $S_n$ for $n\ge3$. To do this, if I define some isomorphic function say $\phi$, where $\phi: S_n \to{\rm Inn}(S_n)$, then show that $\phi$ is bijective (by ...
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### A question on group isomorphism between $(\Bbb{R},+)$ and $(\Bbb{C}, +)$ .

$(\Bbb{R},+)$ and $(\Bbb{C}, +)$ are group-isomorphic. Consider, two vector space $\Bbb{R}_{\Bbb{Q}}$ and $\Bbb{C}_{\Bbb{Q}}$. Any two vector space over the same field having same dimension are ...
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### Prove or disprove : i) $(\mathbf{S},+,.)\cong (\mathbf{R},+,.)$ (as rings) ii) $(\mathbf{S},+)\cong (\mathbf{R},+)$ (as groups)
Let $\mathbf{S}=\left\{\left[\begin{array}{ll}\mathbf{a} & \mathbf{b} \\ 0 & \mathbf{a}\end{array}\right]: \mathbf{a}, \mathbf{b} \in \mathbb{Z}_{2}\right\}$, \$\mathbf{R}=\mathbb{Z}_{2} \times ...