A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Proving $core_G(H)$ is non-trivial.

Question: Let $G$ be a group of order $n^2$, and $H$ a subgroup of order $n$. prove that $H$ contains a non-trivial normal subgroup of $G$. Remarks: -It's equivalent to prove that $\cap_{x\in G} H^...
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6 views

Clairification on Plane Symmetries of a Chessboard: Rotation vrs Reflection

I just wanted to confirm this with someone. When we refer to reflections of the chessboard about the diagonal, we are not talking about rotations (which would assume that the chessboard has the same ...
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2answers
20 views

Lagrange theorem question

I'm trying to teach myself group theory and this question is the final one in an exercise on Lagrange theorem and it has me currently stumped. Finite group ${G}$ contains distinct elements ${a}$ ...
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1answer
22 views

Lagrange's theorem question - have I got it correct?

{${G,*}$} is a group of order 15 with identity element ${e}$. There is an element ${a\in G}$ such that ${a^3\neq e}$ and ${a^5\neq e}$. Prove that {${G,*}$} is a cyclic group with generator ${a}$. ...
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0answers
25 views

A question about unipotent characters

I have just started learning (complex) character theory of groups of Lie type and there are some misunderstandings for me in the subject. It is well-known that unipotent characters of $PGL_{n_i}(q_i)$ ...
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1answer
19 views

Elements a1, … , an are drawn from n distinct left cosets of a subgroup G, each from a different one. …

[question continued...] Show that {${a_1^{-1},a_2^{-1},a_3^{-1} ... , a_n^{-1}}$} is a set of elements, each from a different right coset of G. Find the picture of the question itself here. I get a)...
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2answers
57 views

Corollary of Schur's Lemma - why abelian

Corollary (of Schur's Lemma): Every irreducible complex representation of a finite abelian group G is one-dimensional. My question is now, why has the group to be abelian? As far as I know, we want ...
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1answer
54 views

“algebraic” literature [on hold]

could anyone recommend me an introduction into (i) group theory, (ii) algebraic number theory? (Consider that as two different questions.) Why do you think, that a book you suggest, is good? Thank ...
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1answer
85 views

Proving $x\in\text{SL}(n,\mathbb Q)$ given finite indices of $x^{-1}Gx$ in $G$ and $x^{-1}Gx$

Denote $G=\text{SL}(n,\mathbb Z)$ and let $x\in \text{SL}(n,\mathbb R)$ such that $$[G:x^{-1}Gx\cap G],[x^{-1}Gx:x^{-1}Gx\cap G]<\infty.$$ Prove that $x\in\text{SL}(n,\mathbb Q)$. I know that $\...
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1answer
36 views

Retractions and Isomorphisms of Fundamental Groups

Suppose there is a retraction from $$S^1 \times D^2 \to S^1 \times S^1.$$ Does that then induce an isomorphism $$\pi_1(S^1) \times \pi_1(D^2) \cong \pi_1(S^1) \times \pi_1(S^1)?$$ Which is obviously ...
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34 views

Is the group of units in $\mathcal{O}_{\mathbb{Q}(\sqrt{2})}$ finite and cyclic?

Let $K=\mathbb{Q}(\sqrt{2})$ and let $\mathcal{O}_K$ be its ring of integers. Consider the group of units in $\mathcal{O}_K$. Is it finite? Is it cyclic? My thought: For any $\alpha = a+b\sqrt{2}$, ...
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2answers
44 views

If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)

The following is from Hungerford's graduate algebra book, Chapter 1, Section 4, Proposition 8. The proposition states that: If $H$ and $K$ are subgroups of a group $G$, then $[H:H \cap K] \leq [G:K]...
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3answers
75 views

Prove that the subgroup $\langle a, b \rangle$ has order 6

Let $a = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ and $b = \begin{pmatrix}0 & 1 \\ -1 & -1\end{pmatrix}$ be elements of $\mathrm{GL}_2\left(\mathbb{Q}\right)$. Prove that the ...
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26 views

Conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z}) × GL_m(\mathbb{Z}/q\mathbb{Z})$ [on hold]

Find the number of conjugacy-classes in $GL_n(\mathbb{Z}/p\mathbb{Z})× GL_m(\mathbb{Z}/q\mathbb{Z})$ of cyclic subgroups of order pq?.
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46 views

Group Theory of Friends [on hold]

How group theory applies on five friends. friends = {f1, f2, f3, f4, f5} f1 and f5 form subgroup. f2 and f3 are anti. who is the identity? and how association works. What example can be set to ...
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71 views

A question about automorphism groups of finite groups

I’ve encountered the following question whilst helping a colleague study for comprehensive exams, and I’m stuck on it: Let $G$ be a finite group such that the natural action of the automorphism ...
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0answers
31 views

What is conjugacy class of group of motion of the plane?

This is a question from Artin. I'm unable to even begin to solve it. I don't understand what does it mean? Do we have to find conjugacy class of $D_n$ ? If so, then how?
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42 views

Decomposition of a group into disjoint subgroups

This is an old exam question and I have no idea how to answer it: Can a group be decomposed into a disjoint union of subgroups?
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1answer
24 views

Is there a standard notion of a group “biaction”?

For a group $G$ there is a natural notion of left action on subsets of $G$ given by $g \triangleright H := gH$. But simultaneously, there is a natural right action as well: $H \triangleleft g := Hg$. ...
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1answer
36 views

About the notation of composition of permutations in Lang's book

In Lang's "Algebra", p.30-31, I'm confused about the order of reading the composition of two permutations. In p.30, it seems that we read it from left to right (see the bottom equations), but for p.31,...
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3answers
54 views

How to determine if certain operation is associative based on Cayley table

I have the following table and I don't know how to determine if an operation is associative based on the table. Is there an easy way to do it? Or it's just brute force \begin{array}{|c|c|c|c|c|c|} \...
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55 views

Show that $H$ is not a subgroup of the group $G=\mathbb{Z}_{24}$ (with operation +)

Let $G=\{0,1,2,...,23\}$, and I know that a subgroup is closed underneath the operation and has the identity element. So in order to show it isn't a subgroup, do I just have to show that $$H=\{a + 24\...
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1answer
17 views

If $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$

Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$. My answer Let $\sigma \...
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1answer
21 views

Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
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1answer
56 views

Every group of order $567$ has normal subgroup of order $27$.

Prove that every group of order $567$ has a normal subgroup of order $27$. Let $G$ be such a group. Then $|G| = 3^4\cdot7.$ Let $H\in\text{Syl}_3(G).$ From the Sylow theorems, we have that $n_3 | 7, ...
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36 views

Conjugacy-classes in $\operatorname{GL}_n(\mathbb{Z}/p^m\mathbb{Z})$ [duplicate]

Enumerate the number of conjugacy classes in $\operatorname{GL}_n(\mathbb{Z}/p^m\mathbb{Z})$ of cyclic subgroups of order $p$?.
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1answer
15 views

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite?

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite? Consider for instance a finite sequence of moves (...
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3answers
59 views

Every permutation is a product of two permutations of order 2

I am trying to solve a problem, not for homework, and it has me stomped! For $n\geq 4$ and $\alpha\in S_n$, $$\alpha=\dot{\alpha}\dot{\beta}$$ where $\dot{\alpha},\dot{\beta}$ are of order 2. I know ...
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29 views

$|G|=p^kq$, $p,q$ prime, $p^k\leq 2q$ implies one of the Sylow subgroups is normal

I could use some help with this one: Let $G$ be a group, $|G|=p^kq$ where $p,q$ are prime and $p^k\leq 2q$. Prove that at least one of the Sylow subgroups is normal.
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1answer
10 views

Group von Neumann algebras

I have a question about group von Neumann algebras structure. If $L(G)$ is a subset of $L(H)$, can we find a subgroup $G_1$ of $H$ such that $L(G_1)$ is isomorphic to $L(G)$? I appreciate any help.
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1answer
48 views

Group of order 175 is Abelian

Question: Prove that any group of order 175 is Abelian. The solution: I am unable to understand why the intersection of the normal subgroups is the trivial intersection. Any help is ...
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1answer
47 views

Why is there no retraction between $D^2 \times S^1$ and $S^1\times S^1$?

Why is there no retraction between $D^1 \times S^1$ and $S^1\times S^1$? I have no idea how to prove. I just know that if there is then $\mathbb{Z}\times \mathbb{Z}$ contains a copy of $\mathbb{Z}$. ...
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43 views

A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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$R$ be a commutative unital ring , is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
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41 views

$\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
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1answer
28 views

Homomorphism between S4 and A4

I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H. I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 ...
5
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0answers
73 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
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33 views

Pronormal subgroups of direct products

Suppose that $G = A \times B$. Let $U = A \times \pi_B(U) \leq G$ such that $\pi_B(U)$ is pronormal in $B$. Then $U$ is pronormal in $G$. This is part of a proof of Proposition 4.3 in Pronormal ...
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Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
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1answer
23 views

possible cyclic group from fundamental theorem of finite abelian

Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}...
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30 views

Isomorphism between semidirect products

Alperin and Bell, Groups and Representations, section 2, Proposition 11, p. 23, states: "Let $H$ be a cyclic group and let $N$ be an arbitrary group. If $\varphi$ and $\psi$ are monomorphisms from $H$ ...
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48 views

Does the converse of Lagrange's theorem hold for any finite Dedekind group?

I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if $d$ divides $|G|$ there exists a subgroup of order $d$) holds for any ...
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3answers
69 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
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35 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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24 views

How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
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2answers
27 views

Normal closure of a subgroup of a free group.

Let $G$ be a finitely generated free (nonabelian) group, $H$ a subgroup generated by some of the generators of $G$, and $a: G\to AG$ be the projection to the abelianization $AG:=G/[G,G]$. Is it true ...
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4answers
76 views

If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
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4answers
54 views

$\mathbb{C}/\mathbb{Z}$ is isomorphic to multiplicative group $\mathbb{C}\setminus\{0\}$ [duplicate]

I have to show that $\mathbb{C}/\mathbb{Z}$ is isomorphic to the multiplicative group $\mathbb{C} \setminus \{0\}$. Proof. Let $f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}/\mathbb{Z}$ be the ...
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1answer
45 views

When is $(\Bbb Z/n\Bbb Z)^\times$ cyclic? [duplicate]

Is the group of units $(\Bbb Z/n\Bbb Z)^\times$ always cyclic? Do we need that $n$ is a prime or something?
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1answer
23 views

All homomorphisms from $Z/4Z$ to $Z/6Z$

I am asked to find all group homomorphisms from $Z/4Z$ to $Z/6Z$. Let $f:Z/4Z \rightarrow Z/6Z$ be such a homomorphism. By definition we have $f(1) = 1$ and therefore $f(0)=f(1 * 0) = f(1) * f(0) = ...