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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Number of non-isomorphic abelian groups of order $1440$, containing an element of order $180$.
I have already shown that there are $14$ abelian groups of order $1440$, up to isomorphism. Now I want to determine how many of those $14$ abelian groups contain an element of order $180$.
I figured ...
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Question about isomorphisms between vector spaces [closed]
Sorry if this post is hard to read because I didn't use latex notation.
I don't know how to use it.
This post assumes AoC.
I opened a new question because I thought it would be better this way that to ...
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prove $U(m) \simeq U(n_1) \oplus U(n_2) \oplus ...\oplus U(n_k)$ using internal direct products
If $m=n_1n_2...n_k$ where gcd$(n_i,n_j)=1$ for $i \neq j$, then $$U(m)=U_{m/n_1}(m) \times U_{m/n_2}(m) \times ... \times U_{m/n_k}(m)$$. $$\simeq U(n_1) \oplus U(n_2) \oplus ...\oplus U(n_k) $$
$\...
user1489347
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Unit sphere isomorphism with SO(3)/SO(2)
In group theory we have the following theorem:
$$\textbf{Th: } \text{ Let $M$ be a homogeneus space of the group $G$ (Has only one orbit under G-action) and $p_0\in M$ be fixed}\\ \text{Let also $H:= ...
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Are the groups $\mathbb{{Z}_2}\times\mathbb{{Z}_{12}}$ and $\mathbb{{Z}_4}\times\mathbb{{Z}_6}$ isomorphic?Why? [closed]
Basically,I know what isomorphism is and about the direct products of groups. But I couldn't able to understand how to tell that two direct product groups are isomorphic to each other even though I ...
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Isomorphism between simple group of order 168 and PSL(2,7)
In our group theory classes at the university, we devoted several seminars to derive the uniqueness (up to isomorphism) of a simple group of order $168$. However, i am stuck on last steps, that prove ...
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Which group is isomorphic to $\mathbb{Q^*} /\mathbb{Q^*}^2$?
Let $(\mathbb{Q}^* ,×) $ be the multiplicative group of non-zero rationals and $(\mathbb{Q^*}^2,×)$ be the subgroup formed by squares of rationals.
Which group is isomorphic to $\mathbb{Q^*} /\mathbb{...
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Easy (graphic) interpretation of the third isomorphism theorem of groups
I came across these slides today and this diagram is shown when the third isomorphism theorem was presented.
Could somebody please provide an intuitive way to explain the picture? Why do the green ...
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Subgroup of a finitely generated abelian group
Let $G=\mathbb{Z}\times \mathbb{Z}/10\mathbb{Z}\times \mathbb{Z}/100\mathbb{Z}$. Let $H$ be the subgroup generated by the elements $(1,1,1)$ and $(1,2,3)$.
(1) What is the isomorphism class of $H$? $\...
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how $(-1)^a(5)^b$ produce all elements in $U(2^n)$
I want to prove that $U(2^n) \simeq \mathbb Z_2 \oplus \mathbb Z_{2^{n-2}}$ for $n \geq3$.
I found this in the site.
The isomorphism is shown by $(-1)^a(5)^b \to (a,b)$ such that $(a,b) \in \mathbb ...
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general version of $Z^*_{st} \cong Z^*_{s} \oplus Z^*_{t}$
i am new in the site. Next week, i will take final exam on group theory. Today i wached i video on youtube: https://www.youtube.com/watch?v=QOxyxsCdkz0
In video, it is proved that $Z^*_{st} \cong Z^*_{...
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What is the smallest $m$ such that $\operatorname{SL}_n(q)$ embeds in $S_m$?
The Question:
What is the smallest $m$ (dependent upon $n,q$) such that $$\operatorname{SL}_n(q)\hookrightarrow S_m?$$ That is, such that $\operatorname{SL}_n(q)$ embeds in $S_m$.
Context:
I want to ...
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Prove it using induction :$U(m) \simeq U(n_1) \times U(n_2) \times \cdots \times U(n_k)$
My question is derived from this paper, page $13$ exercise $6.1.37$
Let $m=n_1n_2\cdots n_k$ where gcd$(n_i,n_j)=1$ for $i \neq j$. Then $$U(m) \simeq U(n_1) \times U(n_2) \times \cdots \times U(n_k)....
user1380740
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Construction of groups $G$ such that $G$ is isomorphic to $G×G×G$. [closed]
Construction of (infinitely many) groups $G$ such that $G$ is isomorphic to $G×G×G$.
I know that any infinite dimensional vector space will work here. But apart from that, is there any construction ...
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Show that $G_1 \times G_2 \times \cdots \times G_n \simeq G_{\sigma(1)} \times G_{\sigma(2)} \times \cdots \times G_{\sigma(n)}$
For any $n$ groups $G_1,G_2,\ldots,G_n$ and a permutation $\sigma \in S_{n}$ .Then $$G_1 \times G_2 \times \cdots \times G_n \simeq G_{\sigma(1)} \times G_{\sigma(2)} \times \cdots \times G_{\sigma(n)...
user1403247
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For any subset $\{i_1,…,i_k\} \subseteq \{1,…,n\}$,. the group $G_1×⋯×G_n$ contains a subgroup isomorphic to $G_{i_1}×⋯×G_{i_k}$
My question is derived from one comment in this question. The comment by @ArturoMagidin, says that
For any subset $\{i_1,…,i_k\} \subseteq \{1,…,n\}$,. the group $G_1×⋯×G_n$ contains a subgroup ...
user1380740
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Cyclic groups isomorphic as rings
I know that $\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m \mathbb{Z}$ and $\mathbb{Z}/nm \mathbb{Z}$ are isomorphic as rings, if and only if, they are isomorphic as groups, which happens exactly when $\...
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Why is U6(2) a subgroup of the Conway group
The symmetry group of the Leech lattice is the double cover of Conway's sporadic group $Co_1$. The shortest nonzero vectors in this lattice are length $2$, and the stabilizer of such a vector is ...
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Amalgamated Product Isomorphism
Let $H, G_1, G_2$ be groups such that there exists homomorphisms $f_1:H \to G_1$ and $f_2:H \to G_2$. Furthermore let $H \simeq G_1 *_H G_2$. Does it follow that $H \simeq G_1, G_2$?
I am particularly ...
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Basic group theory, Kernel and Coker of $f' : A/B\to C/D$
Let $A,B,C,D$ be abelian groups.
Suppose $B$ is a subgroup of $A$ and $D$ is a subgroup of $C$. Let $f : A \to C$ be a group homomorphism such that $f(B) \subseteq D$. Then $f$ induces a group ...
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explicit isomorphism from $(\mathbb{Z}/p\mathbb{Z})^{\times}$ to $Z_{p-1}$ [duplicate]
It is well known that the finite subgroups of the multiplicative subgroup $F^{\times}$ of a field $F$ is cyclic.
For the field $\mathbb{F}_p$, its multiplicative subgroup $(\mathbb{Z}/p\mathbb{Z})^{\...
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Proving if K is a subgroup of H, then $\phi^{-1}(K)=\{g \in G :\phi(g) \in K \}$ is a subgroup of $G$ [duplicate]
Suppose that $\phi$ is an isomorphism from a group $(G,.)$ onto a group $(H,*)$.
Then, if K is a subgroup of H, then $\phi^{-1}(K)=\{g \in G :\phi(g) \in K \}$ is a subgroup of $G$
I know that if $\...
user1373312
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A quick way to verify whether two semidirect products are isomorphic.
This question arises from my classifying groups $G$ of order $4r$ where $r\geqslant5$ is a prime. Let $r=\begin{cases}p\equiv1\ (\mathrm{mod}\ 4)\\q\equiv3\ (\mathrm{mod}\ 4)\end{cases}$. Let $T\in\...
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Prove $C_2\times C_3$ is isomorphic to $C_6$ [duplicate]
I am struggling to understand whether or not this proof is legitimate, here is my attempt:
Prove that C2 × C3 is isomorphic to C6
Define $\varphi:C_6\rightarrow C_2\times C_3$ by the mapping $\varphi(...
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Isomorphism for complex multiplication on the unit circle and addition modulo $2 \pi$ on $[0,2 \pi)$
I'm a very beginner in Algebra trying to understand an isomorphism example.
Consider complex multiplication on the unit circle U where $|z| =1$ and addition modulo $2 \pi$ on $\Bbb R_{2\pi} := [ 0,2 \...
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Isomorphism Between the Group of Isometries of an Equilateral Triangle and $( S_3 )$
Let $G$ be the group of isometries of the Euclidean affine plane that leave an equilateral triangle $∆$ invariant. I need help proving that $G$ is isomorphic to $( S_3 )$.
$S_3$ denotes the group of ...
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The Uniqueness of the Logarithm as a Group Isomorphism between the Positive Reals and Reals
My Group Theory textbook asks of me that I prove the following statement:
If $\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}$ be any isomorphism between the groups $(\mathbb{R}^{+}, \cdot)$, $(\...
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Show That For Every Finite Group $G$ and Field $𝔽$ - $G$ is Isomorphic to a Sub-Group of $GL(V)$ for Some finite-dimensional Vector Space $V$ over F
I need to show that if a group $G$ is finite, then for every field $𝔽$ there's a finite-dimensional vector space $V$ over it such that $G$ is isomorphic to a sub-group of $GL(V)$.
I figured the ...
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Describe all non-isomorphic groups of order $57$
Describe all non-isomorphic groups of order $57$, such that for each of them you write down its generators and the connections between them.
Attempt: 57 is the product of two primes, specifically $57 =...
user1345542
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Does there exist a group $G$ such that $\operatorname{Aut}(G)\cong D_5$, where $D_5$ denotes the dihedral group of order 10?
I came across this problem stated in the title having no clue what to do, and got stuck even in the finite case. Here's my attempt:
I first proved that the group $G$, if it exists, cannot be finite ...
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If $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian.
I want to show the following two statements:
a) Let $G$ be a group such that $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian.
For part a) I have the following facts "If $G/...
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Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.
Problem: Show that if $\mathbb{L}(\mathbb{R}^2)$, the space of the linear functionals in $\mathbb{R}^2$, with the matrix norm $\|\cdot\|_p$ ($p > 1$) is isomorphic with the $\mathbb{R}^2$ space ...
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Extending bijection on generators to isomorphism of groups
Given two group presentations $G=\langle X \vert R\rangle$ and $G=\langle Y \vert S\rangle$, let $f:X\to Y$ be a set function only on the generators such that $f(x_1)\ldots f(x_n)=1$ for $x_1\ldots ...
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Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$? ("Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki)
I am reading "Logic in Mathematics and Set Theory" by Kazuyuki Tanaka and Toshio Suzuki.
Problem 1.17
Is $(\mathbb{Q},0,+,<)$ isomorphic to $(\mathbb{Q}^{+},1,\cdot,<)$?
My attempt:
...
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Prove that $S_4/K \cong S_3$ using the fundamental theorem on homomorphism.
Let $A$ be the set formed by the elements of Klein group but identity, A= { (1,2)(3,4);(1,3)(2,4);(1,4)(2,3)}. Consider the set $Big(A)$ of bijection from A to itseld. With the operation of ...
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Natural map of automorphism groups
Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
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Use of correspondence theorem for groups to prove that $o(N) = 2$
Let $G$ be a group and $H \triangleleft G$ simple such that $[G : H] = 2$. I have to prove that if $N \neq \{1\}$, $N \triangleleft G$ and $N \cap H = \{1\}$ then $o(N) = 2$.
I know by third ...
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Screwing up basic short exact sequence $0\to\mathbb Z\leftrightarrow\mathbb Z\to0$
Represent an isomorphism by $\leftrightarrow$.
HAVE exact sequence.
$$ 0 \rightarrow \mathbb Z \leftrightarrow \mathbb Z \rightarrow 0 $$
Then
$$ \text{img} \left( 0 \rightarrow \mathbb Z \right) = 0 =...
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Isomorphism of topological groups
Question: Let $\mathbb{Q}(\sqrt{\mathbb{Q}})$ be the subfield of $\overline{\mathbb{Q}}$ generated by $\{\sqrt{x} : x \in \mathbb{Q}\}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\sqrt{\mathbb{Q}})$ ...
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Commutativity of the wreath product
Let $G$ be a subgroup of the symmetric group $\mathfrak{S}_n$ and $H$ be a subgroup of $\mathfrak{S}_m$. Recall that the wreath product $G \wr H$ is the semi-direct product $G^m \rtimes H$, where $H$ ...
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Computing whether two finite groups are isomorphic (in C++) [closed]
I need to algorithmically compute whether two given finite groups are isomorphic. Usually I only have generators of these groups.
The groups can get quite large as I'm working with subgroups of $S_{32}...
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For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. Compute subsets of $X$ and $Y$ that generate isomorphic subgroups of $G$.
For some finite group $G$ assume $X,Y \subset G, X \cap Y = \emptyset$. I need to compute subsets $X' \subset X$ and $Y' \subset Y$ that generate isomorphic subgroups of $G$: $\langle X' \rangle \leq ...
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Let $A=\langle\{a_1,\dots,a_k\}\rangle,B=\langle\{b_1,\dots,b_k\}\rangle(ord(a_i)=ord(b_i), i=1,\dots,k)$. If $|A| = |B|$, are $A$ and $B$ isomorphic?
Let $A=\langle\{a_1,\dots,a_k\}\rangle,B=\langle\{b_1,\dots,b_k\}\rangle$ where $a_i$ and $b_i$ are elements of some group and $ord(a_i)=ord(b_i), i=1,\dots,k$. In general, if $|A| = |B|$, are the ...
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For $M,C\subset S_n,|M|=|C|$ find subsets of $M$ and $C$ that generate isomorphic subgroups of $S_n$ and the isomorphism maps these subsets together
I have sets $M,C \subset S_n$, s.t. $|M| = |C| \gg 1$. Given $a \in S_n$ I can determine whether $a \in M$ but I have no way to determine whether $a \in C$, however, if we assume C is numbered for ...
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Is a Rational Map Possible Between Elliptic Curves of Orders 19 and 18?
I'm exploring the relationship between elliptic curves and their orders. I have two elliptic curves:
E1: An elliptic curve with order 19.
E2: An elliptic curve with order 18.
Since the order of E1 is ...
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A question on an isomorphism between ${\rm PSL}_2(9)$ and $A_6$ [closed]
I found a nice argument proving that ${\rm PSL}_2(9)\cong A_6$ on page 52 of The finite simple groups by Prof. R.A. Wilson.
Let $f:A_6\to S_{{\rm PL}(9)}$ with $(123)^f= z\mapsto z+1$, $(456)^f= z\...
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Question over decomposition of $\mathbb{Z}_{mn}$
If $m<n\in\mathbb{N}$ and $(m,n)=1$, then there is a natural isomorphism $h: \mathbb{Z}_{mn}\to \mathbb{Z}_m \times \mathbb{Z}_n$. But I'm a little confused about what happens when multiplying $m$ ...
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Every abelian quotient group is a quotient group of $G/G'$ [duplicate]
A group theory book (not in English) I'm reading states the following:
$G/G'$ is the largest abelian quotient group. In fact, every other abelian quotient group is also a quotient group of $G/G'$, ...
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Determining if the three-equation model is isomorphic to the three-point plane model
I'm self studying off of the textbook "Axiomatic Geometry" by John Lee. During the text, the author mentions of the following models of incidence geometry:
Three-point plane model
(1) A &...
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proving that $A_n/\operatorname{ker} \beta \cong \operatorname{ker} \alpha_{n - 1}.$
Prove that in any exact sequence $$\dots \xrightarrow{\alpha_{n+3}}A_{n+2} \xrightarrow{\alpha_{n+2}} A_{n+1} \xrightarrow{\alpha_{n+1}} A_{n} \xrightarrow{\alpha_{n}} A_{n -1}\xrightarrow{\alpha_{n-...
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