Questions tagged [group-theory]

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 is the study of groups.

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

Permutations and Internal direct product.

Suppose $X=Y_1\cup Y_2\,\cup \dots \cup\, Y_n$ where the $Y_i$s are disjoint. In the notes provided by the professor he mentioned that the symmetric group $\operatorname{Sym}(Y_i)$ can be viewed as a ...
1
vote
0answers
26 views

is $p$ is prime, then the nonzero elements of $ℤ_p$ form a group of order $p-1$ multiplication

I defined $ℤ^*_p = \big\{\overline{a}∈ℤ _p\vert \overline{a}≠0\big\}$.The order of $ℤ_p$ is $p$ so $ℤ^*_p $ has order $p-1$ since it excludes only one element.The smallest prime is $2$ thus each $ℤ^*...
1
vote
1answer
40 views

Group homomorphisms on finite groups

Suppose there exists a homomorphism from a finite group $G$ onto $G'$, and that $G'$ has an element of order $n$. How do we prove that $G$ has an element of order $n$? (From Lagrange's Theorem, we ...
1
vote
1answer
44 views

Showing $A= A_r \bigoplus A_s$

I'm trying to answer the following question from Lang's Undergraduate Algebra: I first attempted to prove this directly, however I failed at proving uniqueness. After my first attempted, I used an ...
0
votes
2answers
60 views

Group and ring isomorphism [on hold]

Why is the group $2\mathbb Z$ (group of even integers) under addition, group-isomorphic to the group $\mathbb Z$ under addition, whereas, the ring $2\mathbb Z$ is not ring-isomorphic to the ring $\...
4
votes
2answers
55 views

Group action of $S_n$ on 2n elements [on hold]

Is there a transitive group action of $S_n$, $n > 4$ on a set with $2n$, respectively $4n$, elements?
0
votes
1answer
23 views

Count degrees of freedom symmetric tensor

How do I count the numer of degrees of freedom in a totally symmetric tensor with $m$ lower (or upper) indices taking values in $\{ 1,2,3 \}$? I know that the result is $$dim(m,0)=\frac{(m+2)(m+1)}{...
1
vote
1answer
51 views

Normal subgroup and ideal inclusion

I'm a student who just started abstract algebra. There is an easy question for you. Let $X_1 \le Y$ which means $X_1$ is a subobject(subring or subgroup) of $Y$ Also let $X_2$ bes a (normal or ...
1
vote
0answers
22 views

Decomposition of Infinitely Generated Abelian Groups [duplicate]

I am curious if there is any structure result for an infinitely generated abelian group $G$? In particular, is the following naive guess true? $$G\cong \bigoplus_{i\in A}\mathbb{Z}\oplus \bigoplus_{...
1
vote
1answer
30 views

Maximal subgroups of $p$-groups

The following is exercise 27 from section 6.1 in Dummit and Foote (3rd edition): Let $P$ be a $p$-group and let $\overline{P} = P/frat(P)$ be elementary abelian of order $p^r$. Prove that $P$ has ...
0
votes
1answer
50 views

Simple proof that the order of an element of a group divides order of the group itself

The following theorem have been given to us: "Let $G$ be a group and $a \in G$ be an element of it. Let $k \in \mathbb Z^+$ be a positive integer. Suppose that $|a|=n$. Then $<a^k>=<a^{gcd(n,...
4
votes
0answers
45 views

Overlap of left cosets and right cosets

Let $G$ be a group, and let $H$ be a subgroup of finite index $r$. Let $\{\alpha_i\}$ and $\{\beta_i\}$ be complete sets of right and left coset representatives, respectively. Then, \begin{align*} G= \...
2
votes
2answers
31 views

Characterising maximal subgroups of cyclic groups.

Suppose $G = \langle x \rangle $ is a cyclic group of order $n\geq 1$. Prove that a subgroup $H \leq G$ is maximal if and only if $H = \langle x^p \rangle$ for some prime $p$ dividing $n$. Source ...
1
vote
1answer
28 views

Orbit type and conjugacy class

I am studying an article on symmetric configurations and I found a statement about group theory that I can not understand. Definition: Each orbit is associated a conjugacy class of subgroups of $G$, ...
0
votes
0answers
38 views

Infinite orders (group theory) as cardinals

Let be $(G,\cdot)$ an infinite group. Usually, the order of an element $a\in G$ defined as the smallest positive integer such that $a^n=1.$ Background: The equality $[G:K]=[G:H][H:K]$ for $K\le H\...
2
votes
0answers
38 views

Peter-Weyl Theorem on the Sphere

The Peter-Weyl theorem says that the matrix coefficients of the unitary irreps of a compact topological group $G$ form an orthonormal basis for $L^2(G)$. Similarly, spherical harmonics provide an ...
1
vote
2answers
73 views

Meaning of $\setminus$ notation in Group Theory

Usually $\setminus$ means "set minus", for example $A\setminus B$ means the elements of $A$ not in $B$. But in one of my books on Complex Analysis, we have to show that the automorphisms of the unit ...
4
votes
1answer
37 views

Realizing an isomorphism of (faithful) semidirect products

Suppose that we have two finite groups $A$ and $B$, and suppose that $A$ is abelian. Let $G_{\phi} = A \rtimes_{\phi} B$ be the semidirect product given by $\phi: B \to \operatorname{Aut}(A)$ , and ...
2
votes
0answers
45 views

Abelian Group Criteria and counterexample

I am reading I.N.Herstein-Topics in Algebra I am able to solve the following exercise: If $G$ is a group in which $(a.b)^i=a^i.b^i$ for three consective integers $i$ for all $a,b\in G$, $G$ is ...
1
vote
0answers
34 views

$G$ is a simple group of order $168$. The normalizer of the largest intersection of a pair of Sylow $2$-subgroups is isomorphic to $S_4$.

For more detail, please see this Proposition 2.15. I have two questions about it. First, it's possible that $U\cong \mathbb{Z_2}\times\mathbb{Z_2}$? Second, in the last paragraph of the proof, it ...
0
votes
1answer
48 views

How many homomorphisms $f\colon Z_{99} \to Z_{99}$ are there?

I got this question earlier: How many homomorphisms $f\colon Z_{99} \to Z_{99}$ are there? Which I answered with 100: First you have the identity homomorphism, and all it's shifts ($x\to x+1$, $x\...
0
votes
1answer
27 views

Show that $H\leq A_n$ or $(H: H\cap A_n) = 2$ [on hold]

Let $H$ be a subgroup of $S_n$. Show that $H\leq A_n$ or $(H: H\cap A_n) = 2$. I know that if $$A_n = \{ \alpha \in S_n :\, \alpha \textrm{ is an even permutation} \}$$ so $A_n$ is a subgroup of $...
2
votes
2answers
59 views

The functors $\mathscr M^{op}\to\mathbf{Set}$ are the right $M$-sets

This answer shows that for a one-object category $\mathscr M$ with object $\star$ (corresponding to a monoid $M$), the functors $\mathscr M\to\mathbf{Set}$ are the left $M$-sets. I don't understand ...
-1
votes
1answer
60 views

Artin Algebra 5.7.6 [duplicate]

I was thinking about this problem but could not come up with any normal example. I was trying to crack with some playing with some number theory, like gcd, lcm but could not solve it. I am interested ...
1
vote
1answer
18 views

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable?

Let $G$ be a finite group of automorphisms of $E$ and set $F=Fix(G)$, Then why is $E:F$ always separable ? I have a feeling that it has something to do with the idea that if $E:F$ is separable hence ...
1
vote
1answer
30 views

In which n, do we relate O(n) and Spin(n) to SO(n) via a direct product?

We know that the Lie group (here with real $\mathbb{R}$ coefficients in the matrix representation) --- giving that orthogonal group and spin group are related to the special orthogonal group via $$ 1\...
0
votes
1answer
49 views

The character group

Prove that the one-dimensional characters of a group $G$ form a group under multiplication of functions. This group is called the character of $G$. Here's my try. Let $g$ be an element of order $k$ ...
0
votes
2answers
55 views

What is the kernel of $\exp \colon (\mathbb{R}, +) \rightarrow (\mathbb{R}, \cdot)$? [on hold]

Let $\exp \colon (\mathbb{R}, +) \rightarrow (\mathbb{R}, \cdot)$ be a homomorphism such that $\exp(a+b) = \exp(a)\cdot \exp(b)$. I think the kernel is $\ker(e^x) = \{0, \{-x, x\}\}$. But I want to ...
2
votes
1answer
51 views

Presentations of discrete subgroups of $\textrm{PGL}_2(\mathbb{R})$

It is well known that geometrically finite Fuchsian groups, or finitely generated discrete subgroups of $\textrm{PSL}_2(\mathbb{R})$ can be classified up to isomorphism by their signature $[g,s;m_1,\...
0
votes
0answers
34 views

orbits and decomposition

so I don't understand this clearly. Why are there two orbits with order 5? I understand that rotating a dodecahedron with one face means rotation of other faces too. That is why the rotation group on ...
3
votes
2answers
220 views

Commutator subgroup of Heisenberg group.

Dears, Let $H$ be Heisenberg group, a group of $3\times 3$ matrices with $1$ on the main diagonal, zeros below, and elements of $\Bbb R$ above the main diagonal. Its center is the subgroup of all ...
0
votes
0answers
24 views

Dense projections of lattices in $G\times\textrm{Aut}(T)$

Let $G$ be a simple connected Lie group and let $T$ be a $k$-regular tree. Let $\Gamma$ be a lattice in $G\times\textrm{Aut}(T)$. Assume that the intersection of $\Gamma$ with each direct factor is ...
1
vote
0answers
90 views

Group Theory on Collatz conjecture [on hold]

\begin{matrix} \hline & odd& & &even& &\\ \hline 1&2&3&4&5&6\\ \hline n\equiv 0\left ( mod3 \right ) & n\equiv 1\left ( mod3 \right ) & n\equiv 2\...
1
vote
2answers
52 views

Describe the maximal ideals in the ring of Gaussian Integers $\Bbb Z[i]$. [duplicate]

Describe the maximal ideals in the ring of Gaussian Integers $\Bbb Z[i]$. So first of all my question would be - Is it possible to write any ideal of $\Bbb Z[i]$ (which happens to be a PID) as $\...
5
votes
1answer
70 views

Relationship between $S(G)$, $\text{Aut}(G)$

Let $G$ be a nontrivial group, denote $\text{Aut}(G)$ the group of all its automorphisms of $G$ and denote $S(G)$ the symmetric group on $G$, e.g. the set of all bijections $f:G\rightarrow G$. I would ...
5
votes
0answers
77 views

Is there some sort of classification of invertible finite groups?

Let’s call a group $G$ invertible, if $\forall H \triangleleft G$, there $\exists K \triangleleft G$, such, that $K \cong \frac{G}{H}$ and $H \cong \frac{G}{K}$ All finite abelian groups are known to ...
-6
votes
1answer
24 views

Does The set of two ordered pairs need to be itself ordered for it to be equal to a set of two ordered pairs? [on hold]

$$< <2,7>, <5,7> > ~= ~<<5,7> , <2,7> >~~?$$ Does The set of ordered pairs need to be itself ordered for equivalent term? (For the first notation be equal to the ...
-4
votes
0answers
25 views

I am confused in this [on hold]

Let a group G=D3 and H be any complex which is an abelian subgroup. Is it true that the normalizer of H in G is the whole group G?
3
votes
2answers
114 views

Find the number of the subgroups

Here is the question that I've stuck. Question) $G = Z_{100} \times Z_{500}$ Let $H$ is a subgroup of the $G$ How many number of the $H$ s.t. $H \simeq Z_{25} \times Z_{25}$ ? My idea) This is ...
2
votes
0answers
26 views

Group Theory - product of function periods

Let $G$ be a group and $f : G \rightarrow G$ a function. Prove the set of all periods of $f$ is a subgroup. (A period is any $a \in G$ such that $\forall x \in G, f(ax) = f(x)$) If we let $H$ be the ...
1
vote
0answers
44 views

Is the largest order of an element of a group always the size of the group? [duplicate]

Similar to how one can find a primitive root element of $\mathbb{Z}/p\mathbb{Z}$, I was wondering if, for any group of size $N$ that can be written as $(k\backslash\{0\},\cdot)$ where $k$ is a field, ...
1
vote
0answers
28 views

Reference: set with two coherent group structures.

Do the sets with two group structures which are coherent in some nontrivial sense appear somewhere in mathematics as appropriate structures to investigate? If so what are the areas where they appear ...
1
vote
1answer
21 views

Does the maximum number of roots in a field directly imply the maximum number of solutions in a group

From Proposition 2.5 from https://wstein.org/edu/2007/spring/ent/ent-html/node28.html#prop:dsols, the maximum number of roots $\alpha\in k$ of $x^n-1$ in a field $k$ is $n$. That is, there are at most ...
2
votes
0answers
59 views

Why we say that an arbitrary element of $(Z, +)$ is of infinite order?

Why we say that an arbitrary non-identity element of $(Z, +)$ is of infinite order? I know it is cyclic, because it is possible to express all elements as multiples (repeated additions) of the ...
0
votes
1answer
33 views

How much the discrete subgroups of $\mathrm{O}(n)$ can be complicated?

I have no intuition about the difficulty of classification of discrete subgroups of $\mathrm{O}(n)$ and I wanna to know about it. How much the discrete subgroups of $\mathrm{O}(n)$ can be ...
2
votes
2answers
41 views

Is it necessary for this abelian group to have $4n+2$?

$\DeclareMathOperator{\ord}{ord}$ Let $G$ be a finite abelian group with $|G|=4n+2$, where $n\in \mathbb{N}$. Prove that the product of all of $G$'s elements is different from $e$. I have two ...
0
votes
0answers
23 views

What's the number of all the Sylow subgroups of $GL(4,2)$ [on hold]

Following this post here: Size of conjugacy classes in $GL(4,2)$ How can I further find except the conjugacy classes and its order, the number of all of the Sylow subgroups of $GL(4,2)$?
2
votes
1answer
32 views

Lie Group Homomorphism from $U(n)$ into $SO(2)$

The task was to find an onto Lie Group Homomorphism from $U(n)$ onto $SO(2)$ and I know how to do this. My prof then said that the crux was that the task said onto, if it would have said into you ...
4
votes
0answers
51 views

Is there a way to find the subfields of a Galois extension without knowing the subgroup structure of the corresponding Galois group?

Say we have the polynomial $x^4-2$, the splitting field of this over $\Bbb Q$ is $\Bbb Q(\alpha, i)$, $\alpha=\sqrt[4]{2}$, and its Galois group is isomorphic to $D_8$. Now I know a way to find the ...
5
votes
1answer
47 views

If $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$, then $H$ is normal in $G$.

A normal subgroup $H$ of a group $G$ is said to be a direct factor if there exists a subgroup $K$ of $G$ such that $G\cong H\times K$. If $H$ is a direct factor of $K$ and $K$ is a direct factor of $G$...