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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Proof that the order of any finite $p$-group is a power of $p$

What is the most concise proof that the order of any finite $p$-group is a power of $p$?
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181 views

To show from definitions , if $|G|=15$ then $G$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$

How to show that any group of order $15$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$ ? Please don't use results like "every group of order $15$ is abelian , cyclic " etc. just the definitions . ...
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49 views

Does the binary operation $m ⋆ n = m^n$ on $\mathbb N$ have a neutral element?

Does the binary operation $\,m ⋆ n = m^n\,$ on $\,\mathbb N\,$ have a neutral element? I said yes, and it is $\,e=1\,$ because $\,m ⋆ e = m^e = m^1 = m,\;$ but apparently that is wrong.
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40 views

Cycles of odd length: $\alpha^2=\beta^2 \implies \alpha=\beta$

Let $\alpha$ and $\beta$ be cycles of odd length (not disjoint). Prove that if $\alpha^2=\beta^2$, then $\alpha=\beta$. I need advice on how to approach this. I recognized that $\alpha,\beta$ are ...
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45 views

Orbit and stabaliser of $2\times2 * 2\times1$ matrices

I have the group action of matrix multiplication, meaning: $g((x,y))=\begin{pmatrix}a&0\\0&b\end{pmatrix}$$ \begin{pmatrix}x\\ y\end{pmatrix}$ ...
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77 views

Is orbit a group?

Let a group $G$ act on a set $S$, and let $s$ be an element in $S$. The identity of the orbit of $s$ is $s$ itself and if $a$, $b$ are in the orbit of $s$, then $a b = g_1 g_2 s$, where $g_1$, $g_2$ ...
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66 views

Proving that a set is a group under addition

To show that a set $G$ is a group under addition, do we first need to show that $G$ is closed under addition, or is that implied by proving the three properties of a group, namely there exists an ...
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231 views

Is it true that group of real numbers under addition isomorphic to group of complex numbers under addition [duplicate]

Is it true that set of all real numbers under addition isomorphic to set of all complex numbers under addition
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43 views

Let $H$ be subgroup of $G$, $a,b\in G$, how that $Ha=Hb$ iff $ab^{-1}\in H$.

Let $H$ be subgroup of $G$, $a,b\in G$, how that $Ha=Hb$ iff $ab^{-1}\in H$. Here $Ha$ denote the right coset. Please give me ideas of both direction. Thank you.
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A number to the group cardinality power

Well my question is how is possible this: Consider an element $g\in G$, where $G$ is a finite group, then you have: $g^{|G|}=e$ How can I prove it? Thank you.
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49 views

Constructing homomorphism from a special type of function

Let $f:G \to G$ be a function such that $f(a)f(b)f(c)=f(x)f(y)f(z) , \forall a,b,c,x,y,z \in G$ such that $abc=xyz=e$ ; then is it true that $\exists g\in G$ so that $h:G\to G$ defined as ...
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93 views

Every subgroup of a cyclic group is characteristic (using lattice theory).

I want to show for $n\in\Bbb N$, which is not square-free, that every subgroup of $Z_n=\langle x\;|\;x^n\rangle$ is characteristic. But I want to show it in a convoluted way. Every automorphism ...
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130 views

homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ using $\phi(n)$=$2n$

A question from Visual group theory says : consider the homomorphism $\phi$ : $\mathbb Z$ $\rightarrow$ $\mathbb Z$ by $\phi(n)$=$2n$,where the group operation on $\mathbb Z$ is '+'. Would $\phi$ be ...
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1answer
133 views

Automorphism of $\mathbb{Q}$

How can we show that any automorphism of $\mathbb{Q}$ under addition is of the form $x \to qx$ ,for some q in $\mathbb{Q}$. edited- I found the same question was asked by ...
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118 views

Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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34 views

Onto homorphisms from $S_4$ to $S_2$

Let $S_n$ represent the symmetric group on $n$ letters. How can one find an onto homomorphism from $S_4$ to $S_2$?
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90 views

If $x^2 a x=a^{-1}$, then $a$ has a cube root. [duplicate]

In a group $G$: If $x^2 a x=a^{-1}$, then $a$ has a cube root. (Hint: Show that $xax$ is a cube root of $a^{-1}$.) So essentially $\exists y\in G:a=y^3$. The hint probably confused me more than ...
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60 views

Reflections - Understanding

I have been asked Does the set of all reflections in the plane form a group? Explain. I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I ...
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135 views

Find all group homomorphisms from $\mathbb{Z}^n$ to $ \mathbb{Z}^n$

Can we find all group homomorphism from $\mathbb{Z}^n$ to $\mathbb{Z}^n$? For such a map surjective always imply isomorphism (like $\mathbb{Z}$)?
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69 views

What exactly does $\langle a,b \rangle$ mean? Where a and b are elements of a group G?

Does it mean either a or b will generate the whole of $\langle a,b \rangle$ or does it mean that some of the elements will be generated by a, some by b, and some by $a^rb^q$? The book I'm reading ...
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39 views

$a,b$ are elements of the group $G$. Why $|ba|\leq |ab|$ in the following scenario?

A scenario: The order, $n$, of $ab$ (and hence, by definition, the order of the cyclic subgroup $\langle ab\rangle$) is finite (thus, the order of $ba$ is finite). Then $(ab)^n=e$. So ...
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73 views

A subgroup such that at least one left coset is a right coset

I saw an exercise that goes: Let $G$ be a group of order 120, and $H$ be a subgroup of order 24. If at least one left coset of $H$ in $G$ is a right coset apart from $H$ itself, show that $H$ is ...
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136 views

Must the centralizer of a non-identity element of a group be abelian?

Q1: Must the centralizer of a non-identity element of a group be abelian? I challenge everyone by asking further about this question one step further. The definition of centralizer is: Let a be ...
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59 views

How many recursively definable groups are there on $\mathbb{N}$?

How many non-isomorphic, (non-free), non-trivial, recursively definable groups are there on $\mathbb{N}$? I know we can at least get 1. Let $F:\mathbb{N} \to \mathbb{Z}$ be the "natural bijection". By ...
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60 views

Finitely generated, periodic group such that each conjugacy class is finite must be finite?

This is essentially a repeat of this question. However, the OP didn't seem to put in any work towards a solution and didn't provide context. Nevertheless, I'm still interested in the solution, and I ...
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159 views

Abelianization of the free product of two cyclic groups.

Suppose that $G=G_1*G_2$ where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$ respectively. Show that $G/[G,G]$ has order $mn$. Can anyone help me?
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73 views

How can i prove that a cartesian product is isomorphic to another cartesian product

$\def\<#1>{\left<#1\right>}\def\Z{\mathbb Z}\<\Z_6, \oplus> \times \<\Z_{10},\oplus>$ is isomorphic to $\<\Z_2, \oplus> \times \<\Z_{30}, \oplus>$ i know i have to ...
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70 views

Triviality of $H_3(G,\mathbb{Z})$

We know that the triviality of the Schur multiplier means that projective representations can be lifted to ordinary ones. The Schur multiplier is also a measure of the failure of how the commutator ...
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45 views

Satisfying the hypotheses of the First Isomorphism Theorem.

Let $G$ be the group of invertible upper triangular $2 \times 2$ matrices with real entries and let $H$ be the subset of $G$ consisting of those matrices $A$ with $a_{11} = 1$. It is easy to see that ...
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61 views

Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
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41 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
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1answer
27 views

Normal closure of the nonnormal factor of Holomorph of a Cyclic group

Let $C_n$ be the cyclic group of order $n$. Then, we can consider the holomorph $G=C_n\rtimes Aut(C_n)$. let $H$ be such that $Aut(C_n)\leq H\trianglelefteq G$. Is it necessarily the case that $H$ is ...
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42 views

Simple groups and Schur multiplier

Let $G/K$ be a simple finite non-abelian group. We know that $G/K\cong G^{\prime}/(G^{\prime}\cap K)$. Now I would like to prove $G^{\prime}\cap K$ is a homomorphic image of the Schur multiplier of ...
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1answer
39 views

When $G_1\ast_A G_2=1$?

Is the following possible? $f_1\colon A\to G_1$ and $f_2\colon A\to G_2$ are group homomorphisms and $G_1\ast_A G_2=1$ and neither $f_1$ nor $f_2$ are surjective.
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46 views

Does the Product of conjugates of some subgroups commutes with all elements of another subgroup

Let $U,V \le G$ abelian subgroups such that $V$ is normal and $G$ be finite, does it hold that for $v \in V$ we have $$ u \cdot \left( \prod_{u' \in U} v^{u'} \right) = \left( \prod_{u' \in U} ...
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52 views

Solvable groups of order 25920

I would like to prove the following statement: Let $G$ be a finite solvable group of order $2^6.3^4.5$. If $O_{5^\prime}(G)\neq1$, then $G$ has an element of order $18$. Also, I would like to know ...
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42 views

Are free products of exponential growth?

Let $G$ and $H$ be finitely generated groups. Is the free product $G*H$ always of exponential growth? I am guessing that the answer is yes, but I don't know how to prove it correctly. My idea is to ...
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Why $(\mathbb{Z}/N\mathbb{Z})^{\times}[2]$ is of order $2$?

Why if $(\mathbb{Z}/N\mathbb{Z})^{\times}$ is cyclic, the group of his elements of order dividing $2$ is of order 2?
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100 views

Eulerian path for Rubik's Cube states

There are a number of discussions online confirming that there exists a Hamiltonian cycle through the states of a Rubik's Cube. Or more precisely, the "quarter-turn metric Cayley graph for the Rubik's ...
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1answer
74 views

Do noncyclic groups of order $p^n$, $n>1$ always have a subgroup isomorphic to $C_p\times C_p$?

If $p=2$, then the quaternion group is a counterexammple, so let $p$ be an odd prime. Are there any groups like the quaternion groups for odd primes (i.e., they have a center of order $p$ and every ...
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Cayley graph of an amalgam

I was wondering what the Cayley graph of an amalgam $A*B$ (over $\{ 1\}$ ) is? $A, B$ are finitely generated. Can't work it out.
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What about the converse of this statement?

A subgroup $C$ of a group $G$ is said to be a characteristic subgroup of $G$ if and only if $T[C] \subset C$ for all automorphisms $T$ of $G$. Here $T[C] \colon= \{ T(c) \colon c \in C \}$ is the ...
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75 views

Existence of isomorphism between groups of upper triangular matrices.

Is there an isomorphism between this group of matrices $$ \begin{pmatrix} 1 & k \\ 0 & 1 \\ \end{pmatrix},~~k\in\mathbb Z $$ and this one $$ \begin{pmatrix} 1 & k_1 & k_2 \\ 0 & 1 ...
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127 views

calculate structure automorphism group in MAGMA

Please hint me! Is there any instruction in MAGMA to calculate structure automorphism group by using generators? for example: ...
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56 views

$|p+1|=p^{n-1}$ in $\left( \mathbb{Z}/p^n \mathbb{Z} \right)^\times$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra: Let $p$ be an odd prime and let $n$ be a positive integer. Use the Binomial Theorem to show that $(1+p)^{p^{n-1}} ...
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45 views

Is there a way to encode a ring into a group?

Is there a meaningful bijection between tne set of all rings and the set of all groups? Thanks.
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51 views

Are these subgroups of G only subgroups if G is abelian?

I am doing some exercises in a book I am reading. The exercises and my answers for them are as follows: Let $H$ be a subgroup of $G$, and let $K = \{x \in G: x^2 \in H\}$. Prove that $K$ is a ...
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114 views

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism on $g$, $g(N) =N$.

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism $g$, $g(N) =N$. I can prove this for the case when there is a subgroup $H$ with the same ...
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1answer
42 views

Group actions: orbits equivalent to divisors?

Does there exist a group with a group action, that acts on the set of natural numbers where the orbit of any natural number is the set of its divisors?
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55 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...