# 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 that the given mapping is isomorphism.

Let's define a mapping from A→Ca by f(Ng)=(g^-1)a(g). where: A=set of right cosets i.e.Ng Ca= Conjugacy Class Can anyone explain that how to show this an isomorphism please?
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### Show that he function mapping is isomorphism. [closed]

show that the given mapping is Isomorphism by explaining all steps in details i e. Well Defined, One-One, Onto
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### About isomorphism of two groups. [closed]

If two groups have same number of subgroups of same order , then do they need to be always isomorphic? Ok, that is question which arrived in my mind , when I was going to prove that there is an unique ...
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### Group isomorphism and the structural similarity

A question on group isomorphism. Let (A,・) and (B,*) be some groups, and I want to show they're isomorphic. I know there has to exist a bijection function f: Α → Β such that f(x・y)=f(x)*f(y) for all x,...
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### automorphism, endomorphism, isomorphism, homomorphism within $\mathbb{Z}$

From Wikipedia: An invertible endomorphism of $X$ is called an automorphism. The set of all automorphisms is a subset of $\mathrm{End}(X)$ with a group structure, called the automorphism group of $X$ ...
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### irreducible representations of $\mathbb{Z}_n$

I'm trying to find all the irreducible representations (which are not isomorphic) of $(\mathbb{Z}_n,+)$ in the field of real numbers. I saw that we can realize by mapping a group generator $g$ to $1$ ...
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### Show that $\phi : \mathbb{Z}_2\rightarrow\mathbb{Z}_4^*$ is an isomorphism.

I know that $\mathbb{Z}_2=\{0,1\}$ and $\mathbb{Z}_4^*=\{1,3\}$ So $\phi$ is defined as:$\phi(0)=1$ and $\phi(1)=3$ Clearly then it is bijective since every element of $\mathbb{Z}_2$ maps to exactly ...
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### Let $G$ and $H$ be nontrivial groups such that $G$ is simple and let $f : G \to H$ be a surjective homomorphism. Show that $f$ is an isomorphism. [closed]

Having some discomfort in my solution and was wondering if there was an easier way to do this, Thanks. If $G$ is simple, then the kernel of $f$ is either $\{1\}$ or $G$ itself. Since the kernel is a ...
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### Second isomorphism theorem proof for groups.

I'd like to be prove the second isomorphism theorem. Let $H$ and $K$ be two subgroups of $G$ with $K$ normal in $G$. The subgroup $H \cap K$ is normal in $H$, $HK$ is a subgroup of $G$, and $K$ is ...
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### A group isomorphism between $\mathbb{Q/Z}$ and $\mathbb{Q/2Z}$

Question: Prove that $\mathbb{(Q/Z, +)}\cong\mathbb{(Q/2Z, +)}$ My attempt To prove they are isomorphic I need to define a map from $\mathbb{Q/Z}$ to $\mathbb{Q/2Z}$ which is bijective and preserve ...
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### counter example needed: existence of an isomorphism which maps a “diagonal” (non product) subgroup of a finite abelian group to a product subgroup.

Consider a finite abelian group: $$G \cong \mathbb{Z}/p_1^{\alpha_1}\mathbb{Z} \times\mathbb{Z}/p_2^{\alpha_2}\mathbb{Z} \times\dots\times\mathbb{Z}/p_n^{\alpha_n}\mathbb{Z}$$ Let $K$ be a subgroup ...
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### Twisted subgroups. (Subgroups of $D_5 \times S^1$

Heya all I am looking for twisted subgroups of $D_5$, equivalently subgroups of $D_5 \times S^1$, and was wondering if the group generated by the elements $[\rho,0]$ and $[\kappa,\frac{1}{2}]$ forms ...
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### If $G$ is a group, $H \leq G$, and for $a \in G$, $x \mapsto axa^{-1}$ is an isomorphism. Why isn't this sufficient to conclude $H$ is normal? [closed]

A subgroup $H$ is normal if and only if $\forall a \in G (H = aHa^{-1})$, $a H a^{-1} \subseteq H$ and for arbitrary $a \in G$, $x \mapsto axa^{-1}$ is an isomorphism, why doesn't this imply that $H$ ...
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### Role of semidirect product and intuition for $O(2)\simeq U(1)\rtimes \mathbb{Z}_2$?

I have a very basic understanding of some common groups, and I'm trying to get some intuition for this isomorphism. My thinking so far is that $O(2)$ is rotations and reflections in $\mathbb{R}_2$, ...
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