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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Constructing a group $(G, \Delta)$ isomorphic to $(G, \ast)$. [duplicate]
I encountered this interesting result from the other day and I am struggling to prove it myself.
Proposition: Let $(G, \ast)$ be a group and $\sigma\in \text{Sym}(G)$. Define a binary operation $\...
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How applicable are the isomorphism theorems? [closed]
How broadly do the (four?) isomorphism theorems apply? Do they hold only of groups? What do they look like in set theory?
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First Isomorphism Theorem meaning of "induces" [duplicate]
The first isomorphism theorem states that any homomorphism f from group G to group H induces an isomorphism from G/Ker(f) to the Im(f). Can someone explain what "induces" means here? I see ...
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Proving isomorphism from $\mathbb{Z}_2 \times \pm QR_n$ to $\mathbb{Z}^*_n$ ($QR$ being quadratic residue) for a safe biprime $n$
The statement is, For any safe biprime $n = p \cdot q$ with $p = 2p' +1$ and $q = 2q' +1$, it holds that $\mathbb{Z}^*_n$ is isomorphic to $\mathbb{Z}_2 \times \pm QR_n$, $\pm QR_n$ being the union of ...
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What can we say about groups with isomorphic automorphism groups? [closed]
Given two groups $G$ and $H$ satisfying $$\text{Aut}(G)\cong \text{Aut}(H).$$Are there any interesting facts one can conclude? Does this imply something useful? Can you add additional conditions in ...
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Proof of Diamond Isomorphism Theorem Using Universal Property of Projection
Reading from Categories for the Working Mathematician. The following problem is posed in section III.1:
Use only universality (of projections) to prove the following isomorphisms of group theory:
(a) ...
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If $\phi:M\to P$ is an isomorphism and $A\le M$ a subgroup, then $\phi[A]\cong A$? What if $A=M$ but $\phi$ is only a homomorphism?
If $A$ is a subgroup of a group $M$:
$$A \leq M,$$
if $\phi: M \rightarrow P$ is an isomorphism, then can we conclude that $\phi[A]\cong A$?
(Note, I am using $\cong$ to denote an isomorphism).
Now, ...
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Verification of attempt at finding the quotient group using the first isomorphism theorem
The question is from Artin's Algebra, exercise 2.10.1.
I am unsure mainly of two things: whether the solution is complete and whether everything is fine with regards to identifying the homomorphism ...
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Which small groups appear in multiple classical families like $C_n$, $D_n$, $S_n$, or $A_n$?
On page 90 of Visual Group Theory by Nathan Carter, there's an exercise (Exercise 5.20) that points out an interesting phenomenon: some small groups belong to more than one of the classical group ...
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Need help in continuing the solution of the problem: "Prove that a group of order $108$ has a normal subgroup of order $9$ or $27.$"
Prove that a group of order $108$ has a normal subgroup of order $9$ or $27.$
My solution goes like this:
By Sylow Theorems, we can say that there exists a subgroup of order $27,$ say, $H.$ Suppose $G$...
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Second theorem of isomorphism for groups
Let $G$ be a group, $K$ a subgroup of $G$, and $H$ a normal subgroup of $G$.
I didn't understand why the second theorem of isomorphism is stated as
$$\frac{K}{H\cap K}\cong \frac{KH}{H}$$
instead of ...
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An equivalent definition of isomorphism.
Let $G$ and $\overline G$ be groups of the same order, and $f\colon G\to\overline G$ a bijection. If $ f$ has the
Property. For all $g,h\in G$:
$$f(gh)=f(g)f(1_G)^{-1}f(h)\tag1$$
then $f^*:=f(1_G)^{-1}...
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Must a group isomorphism between $\mathbb{R}$-modules be an $\mathbb{R}$-module isomorphism? [closed]
If $M$, $M'$ are $R$-modules ($R$ is a ring with identity), when $R$ is the set of rational numbers $\mathbb{Q}$, please prove:
If $\eta : M\to M'$ is an isomorphism of abelian groups, then $\eta$ is ...
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Prove that $\phi :(\mathbb{C},+)\longmapsto (\mathbb{C},+)$ is an isomorphism. $\phi$ is define to be $\phi(a+bi)=a-bi$ [closed]
Prove that $\phi :(\mathbb{C},+)\longmapsto (\mathbb{C},+)$ is an isomorphism. $\phi$ is define to be $\phi(a+bi)=a-bi$
I just need a help on starting this one. Here is my attempt
Let $x=a_1+b_1i$ ...
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Non-equivalence of $Q_8$ as a Group Extension of $V_4$ by $C_2$
I've been learning about group extensions and noted the 8 nonequivalent extensions of $V_4$ by $C_2$ listed on wikipedia. I've been trying to figure out what the 8 are and their corresponding groups ...
