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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Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
2
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31 views

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime.

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime where $Z(G)$ is the center of $G$ and $|G:Z(G)|$ is the index of $Z(G)$ in $G$. This was in a test that I had recently but I was not able ...
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1answer
36 views

Prove the following set endowed with the binary operation is an abelian group

Let $∗$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x ∗ z = y$. Show that this set with the operation $∗...
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intersection of all maximal centralizer

Let $G$ be non-abelian. Let say $C_G(x)$ is maximal centralizer if $x\in G-Z(G)$ and there is no $g\in G-Z(G)$ such that $C_G(x)< C_G(g)$. Now let $C$ be the intersection of all maximal ...
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51 views

Does $H_1/\ker(\phi_{|H_1})=H_2/\ker(\phi_{|H_2})$ imply that $H_1=H_2$?

Let $\phi: G \rightarrow G'$ be a group homomorphism that is onto and $H_1 \subseteq H_2$ are subgroups of $G$. In addition, $\phi(H_1)=\phi(H_2)$ and $\ker(\phi_{|H_1})=\ker(\phi_{|H_2})$. I suspect ...
2
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1answer
48 views

Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...
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32 views

Why don't Archimedean solids give finite subgroups of $SO(3,\Bbb R)$?

I know that the Platonic solids correspond to finite subgroups of $SO(3,\Bbb R)$. For example, the tetrahedron corresponds to a subgroup isomorphic to $A_4$. The cube and octahedron to one isomorphic ...
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1answer
22 views

Is my proof of 'If $G$ contains any odd permutations then precisely half of the elements of $G$ are odd' correct?

Let $G$ be a subgroup of the symmetric group $S_n$.Show that if $G$ contains any odd permutations then precisely half of the elements of $G$ are odd. My proof: Let $G$ be a subgroup of $S_n$, and ...
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2answers
52 views

Counter example disproving: if $H = \{g^2 | g\in G\}$ is a subgroup of $G \Rightarrow G$ is abelian.

I am trying to find a counter example to disprove if $H = \{g^2 | g\in G\}$ is a subgroup of $G \Rightarrow G$ is abelian. I can't seem to find one. I suspect it should involve either matrices or ...
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Follow up on when it happens that $b=b^{-1}$

This is a question that came up after my previous question here. In my previous question I asked for help to prove that if $a$ is a group element of odd order and $b$ an element such that $b = a b^{-...
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34 views

If $p^n \mid |G|$ then the finite p-group $G$ has a normal subgroup of order $p^n$

Suppose that $G$ is a finite p-group. Show that if $p^n \mid |G|$ then $G$ has a normal subgroup of order $p^n$. I am stuck on this problem. I know that $G$ has subgroups of order $p^m$ but I don't ...
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1answer
47 views

Group theory puzzle (whirligig?)

This is a problem inspired from the Tomb Raider Anniversary video game (2007?) at the beginning of the Sanctuary of Scion in Egypt when Lara needed to solve the puzzle with 4 columns where the columns ...
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1answer
46 views

What is $\langle a \rangle + \langle b \rangle$ where $a$ and $b$ are natural numbers?

Let $\langle a \rangle$ and $\langle b \rangle$ be ideals in $\mathbb{Z}$, where a and b are natural numbers. Define $$S = \langle a \rangle + \langle b \rangle = \{x + y \mid x \in \langle a \...
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69 views

Prove the identity $\Sigma_{d|n}\phi(d) = n$, where the sum is extended over all the divisors $d$ of $n$.

let $\phi$ denote the Euler's function. Prove the identity $\Sigma_{d|n}\phi(d) = n$, where the sum is extended over all the divisors $d$ of $n$. attempt: Suppose $Z$ is a cyclic group of order $n$. ...
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17 views

Is there any formula with the help of order of a group to check whether group is abalian or not…?

Is there any formula with the help of order of a group to check whether group is abalian or not...?
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1answer
26 views

Abelian group as direct product of its p-Sylow subgroups.

I have read the Sylow 3 theorems, but I don't think I fully understand what they mean. Could someone help clarify them for me. Especially how they might apply to this question. Thanks.
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1answer
43 views

Homomorphism from Rubik's Cube group to Symmetric Group.

$R^2$ is the rotation of the right side by 180, $F^2$ is the rotation of the front by 180. Other than this I am completely lost and this blowing my mind. I wish I had more to build off of, but I am ...
2
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1answer
41 views

Product of 3 non-disjoint 4-cycles from $S_4$

I'm having a lot of difficulty finding the products of permutations and I can't sort out why. If someone could explain to me why $$(1 2 3 4) (1 2 4 3) (1 2 4 3) = (1 3)$$ That would be great.
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1answer
67 views

Characterization of group homomorphisms from $R$ to $R$

I am trying to characterize all the group homomorphisms from $R$ to $R$. I have characterized all the "continuous" group homomorphisms from $R$ to $R$. They are of the form $f(x) = f(1) x$. Now I ...
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1answer
47 views

Isomorphism $\phi:G\to G\times G$ [duplicate]

Can anyone provide me with an example of a non-trivial group $G$ which is isomorphic to $G\times G$. What is the mapping $G \to G\times G$?
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3answers
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What's the angle of rotation of a product of two reflections?

Let $F_1$ and $F_2$ be two arbitrary reflections about two lines in $\mathbb R^2$. I've been trying to work out the angle of rotation of $R_1R_2$. To this end I drew pictures in which I reflect one ...
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78 views

Completing the proof using the fundamental theorem of cyclic groups

$G$ is abelian; $H=${$g\in G,|g| \text { divides } 12$}. Prove that $H$ is a subgroup of $G$. My idea: if the order of $g$ divides 12, then the order of the group $G$ must be some multiple of $12$. ...
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3answers
86 views

Show that $a^{m} a^{n} = a^{m+n}$

Prove that if $G$ is a group and $a\in G$, then we have $\forall m,n\in\mathbb{Z} $ that $$a^{m} a^{n} = a^{m+n}.$$ I've proved the case when $m,n>0$ but I'm stuck on how to prove the case when ...
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2answers
53 views

Relationships between number of conjugacy classes of a group and of a subgroup?

The statement of the problem: Given a finite group $G$ and a subgroup $H$, I need to show that the number of conjugacy classes of $G$, $$ k(G) \leq |G / H|\,k(H) $$ where $k(H)$ is the number of ...
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1answer
54 views

Creating groups with $n$ elements for a given $n$

I would like to create groups with $n$ elements for a given $n$. I know that I can take some product of $\mathbb Z / p_i \mathbb Z$ for some primes $p_i$. But I want to find less obvious (=more ...
2
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1answer
56 views

Finding centralizer of a matrix in general linear group.

I saw the following question from Gallian's book on abstract algebra. I am required to find the centralizer of the matrix $$A= \begin{bmatrix} 1 & 1 \\ 1 & 0 \\ ...
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Outer Automorphisms of PSL2(R)

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would ...
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101 views

non-abelian groups of order $p^2q^2$.

Let $p<q$ be prime numbers and let $G$ be a group of order $p^2q^2$. I wish to determine up to isomorphism how many groups $G$ are there. What I know: The abelian case is very clear. Moreover, ...
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1answer
31 views

Why is this proof incorrect? (Group Theory)

If $G$ is a finite non-trivial group and $H \leq G$ has index $p$ prime, then $H \lhd G$. This statement is actually false. However, I cannot find the error in what I wrote. Let $G$ act on the ...
2
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1answer
39 views

Proving that the multiplicative group modulo $2^r$ are cyclic iff $r<3$

No idea how to start, can someone give me a hint? Show that $(\mathbb{Z}/2^r\mathbb{Z})^*$ is a cyclic group if and only if $r<3$.
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1answer
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$k$ cosets of normal subgroup and $k$th power of group element

I wrote a proof, but I'm afraid it's not quite perfect. If one wants to prove that for a finite group $G$, where $N \trianglelefteq G$, and $|G:N|=k$, then $a^k \in N$ for all $a \in G$, he can ...
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1answer
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Affine linear map, finite-dim. reducible rep. but can't be decomopsed as a direct sum of irreducible subreps?

For $a \in \mathbb{R}^\times$ and $b \in \mathbb{R}$, let$$g_{a, b} : \mathbb{R} \to \mathbb{R}, \text{ }x \mapsto a \cdot x + b$$be an affine linear map. Let$$\text{Aff}(\mathbb{R}) = \{g_{a, b} : a \...
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2answers
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Quotient isomorphism question.

I'm confused about where the equivalence relations come in and how they are related to the maps.
2
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3answers
51 views

Automorphism group for a graph

I don't understand some basic definitions in the setup for the following problem from Springer's Algebraic Combinatorics', and would love some clarification: "For a simple graph Γ with vertex set V , ...
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1answer
124 views

Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads

The question is as stated in the title up to symmetries of $D_{16}$. I know this has to do with the following two formulas: If $G=D_{16}$ is the group acting on the set $S$ of different necklaces, ...
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1answer
73 views

Cyclically reduced words in free groups

Let $F$ be a free group on $\{x_1,x_2,\cdots\}$, $w$ a word in $F$. Then $w$ is a finite expression in $x_i$'s and their inverses. By cancellation, $w$ can be reduced to simplest expression. Let $w=...
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1answer
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Corrolary 15 in *Abstract Algebra* by Dummit and Foote

On page 94, the Corollary states: If $H$ and $K$ are subgroups of $G$ and $H \leq N_G(K)$, then $HK$ is a subgroup of $G$. In particular, if $K \trianglelefteq G$ then $HK \leq G$ for any $H \...
2
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2answers
98 views

Geometric meaning of normal in group theory?

How should one think about normal subgroups intuitively? Is there any useful geometric intuition behind them? For instance, I remember reading somewhere that normal subgroups are like bundles in some ...
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59 views

Commutator Subgroup is finitely generated

It is well known that the commutator subgroup of a finitely generated nilpotent group is finitely generated, in fact all subgroups in this case are finitely generated. I am interested in infinite ...
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Group homomorphism induced by another homomorphism

I need to prove a theorem that if $\phi: G\to H$ is some homomorphism, $M\leq H$ $\phi^{-1}(M) := \{g\in G:\phi(g)\in K\}$, $M \trianglelefteq H$, then $\phi$ induces an injective homomorphism $G/\phi^...
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1answer
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Proving $\gcd(n,\lvert G\rvert) = 1$

Given a finite cyclic group $G$ and any group $H$, let $\phi_1,\phi_2\colon G\to\mathrm{Aut}(H)$ be homomorphisms such that $\sigma\phi_1(G)\sigma^{-1} = \phi_2(G)$ for some $\sigma\in\mathrm{Aut}(H)$....
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2answers
85 views

Subset of a finite group

Let G be a finite group. Let $$H = \{b \in G.\ bab^{−1} \in \langle a \rangle \}.a \in G$$ Prove that if G is a finite group, then H is a subgroup of G.I think that a good approach is to prove that $$...
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2answers
167 views

Group of integer orthogonal matrices

Let $O_n(\mathbb Z)$ be a group of orthogonal matrices B st. $B*B^T=I$ with entries $b_{ij} \in \mathbb Z$. How do I show that $O_n(\mathbb Z)$ is a finite group and find its order. I need to show ...
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1answer
46 views

Sufficient condition to check whether a group is Noetherian

Suppose $G$ is a finitely generated group. What conditions on $G$(or some subgroups of $G$) will force it to be a Noetherian group? Of course, if all subgroups are finitely generated or ACC on ...
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86 views

Equivalent definition of abelian group

Assuming that given operation is commutative and associative, are following definitions equivalent? 1) A zero element $e_0$ exists such that $e_i+e_0=e_i$ and inverse $e_{i'}$ exists for every $e_i$ ...
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Discrete subgroups of $\mathbb R^n$ are free

Let $G$ be a nonzero subgroup of the additive group $\mathbb{R}^n$. Assume $G$ is discrete in the sense that for any $x \in G$, there exists an open set $U \subset \mathbb{R}^n$ such that $U \cap G = \...
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3answers
204 views

Are all abelian subgroups of a dihedral group cyclic?

Are all abelian subgroups of a dihedral group cyclic? Attempt: I have counter-examples for n=1,2 so I know that it isn't true for n<3. Is it true for n≥3? How do you know this?
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2answers
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When is it valid to argue using my 'trivial intersection -> so many distinct elements'?

How do I know if I can make counting statements for sylow theorems that rely on intersecting trivially? I.e it is a common argument that I use in which I rely on we have $x$ amount of sylow $p$-groups ...
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1answer
43 views

What does it mean for a subgroup to fix a vector space?

I was wondering if anyone could explain what is meant by the two statements below? Any help would be greatly appreciated. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of ...
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1answer
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two definition in automorphism of group

Let $G$ be finite p-group and $\sigma \in Aut(G)$ (automorphism group). what does below symbols mean 1. $[G,\sigma]$ (commutator) 2. $C_G(\sigma)$ (centralizer)