The study of symmetry: groups, subgroups, homomorphisms, group actions.

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0
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1answer
16 views

Centralizer of unique cyclic subgroup of order equal to exponent of group 2

I rewrite my question in a better way. Let $G$ be a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|K|=exp(G)$, where $exp(G)=lcm\{|g|\big|g\in G\}$. Is $K=C_{G}(K)$?
4
votes
1answer
22 views

Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
1
vote
1answer
31 views

The Trivial Centre of Group $G/Z(G)$

Let $G$ be a group such that $G = G'$, where $G'= [G,G]$ is the Derived Subgroup of $G$. Prove that the centre o group $G/Z(G)$ is trivial, that is, prove that $Z\left(G/Z(G) \right) = 1$.
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1answer
25 views

Group action of a direct product of groups

Let $G$ be a finite group acting on the $n$-dim vector space $X$. Let $R$ be an $n$-dim representation of $G$. $X$ consists of points $(x1,x2,...,xn)$, which are acted upon by $R(g)$, for $g$ in ...
0
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0answers
25 views

Centralizer of unique cyclic subgroup of order equal to exponent of group [on hold]

Let $G$ a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|N|=\exp(G)$. Is $C_{G}(N)=N$?
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2answers
42 views

If a generated subgroup is cyclic

I would like to make a similar question to question "Exercise on generated subgroup": Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then ...
1
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0answers
21 views

Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
0
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1answer
15 views

semi direct of quaternionic group [duplicate]

Hi I'm having trouble understanding why the quaternionic group cannot be written as a semi-direct product in a non-trivial way. Is there a proof of why this is the case?
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votes
2answers
37 views

Equation in Free Group [on hold]

Let $F$ be a free group and $a \in F$. Assume that for any natural $n>1$ the equation $x^n=a$ has solutions (that is, $a$ is infinitely divisible). Show that $a=1$.
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votes
1answer
44 views

If $ x^2=e$ for all $x\in G$ then $G$ must be [on hold]

If $ x^2=e$ for all $x\in G$ then $G$ must be cyclic non-abelian abelian finite group.
0
votes
0answers
33 views

In a finite group, the equation $x^m=e$ has $m$ solutions for each positive $m$ that divides the order of the group

Show that in a finite group $G$ or order $n$ written multiplicatively the equation $x^m=e$ has $m$ solutions $x\in G$ for each positive $m$ that divides $n$ I am having trouble understanding how ...
0
votes
0answers
32 views

$G$ is the semidirect product of $U_n$ & $D_n$

Let $GL_n \Bbb R$ be the set of all invertible matrices of size n. Let $U_n$ be the set of upper triangular matrices with $1$'s on the diagonal and $D_n$ be the set of diagonal matrices with non-zero ...
0
votes
0answers
44 views

Notes about the ring of $p$-adic integers $\Bbb Z_p$

I'm studing profinte groups. I'm using Wilson's book "Profinite Groups". Here the ring of $p$-adic integers $\Bbb Z_p$ is introduced as inverse limit of rings $\Bbb Z/p^n\Bbb Z$. I'm searching for ...
1
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1answer
29 views

Consequence of Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$

Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$. Theorem to be proved: Suppose G is finite abelian and group of order m, let p be a prime number dividing m. Then G ...
2
votes
1answer
45 views

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...
0
votes
2answers
36 views

Every Two Element in A Coset

For every $a,b \in G$, $a,b \in cH$ for some maximal subgroup $H$ of $G$ and some $c \in G$. For what groups is the following property true? I know its true for $\mathbb{Z_m} \times ...
0
votes
1answer
23 views

Prove that $C_H(K)=N_H(K)$.

Let $H$ & $K$ be groups. Let $\phi$ be a homomorphism from K into $Aut(H)$. Identify $H$ & $K$ as subgroups of $G=H\rtimes_{\phi}K$. Prove that $C_H(K)=N_H(K)$.
0
votes
1answer
18 views

How to find orbits and isoropy group?

About this problem ${a}$, I am wondering if there are 5 orbits in $A$? The 5 orbits separately contain elements which 3 are all the same, 2 of 3 are the same and all 3 are different? I am confused ...
0
votes
2answers
51 views

Group $G$ such that there is a proper subgroup containing every other proper subgroup of $G$

Characterize all the groups $G$with the following property: There is a proper subgroup $H$ of $G$ such that $\forall S$ proper subgroup of $G$, $S \subset H$. I am pretty lost with this exercise. If ...
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0answers
21 views

Integral domain and ordered ring

I know that every ordered ring is an integral domain. Is it true that every ring that is an integral domain is also ordered? By ordered I mean for any $a \in R$, either $a = 0$ or $ a > 0$ or $a ...
2
votes
2answers
62 views

Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
1
vote
2answers
34 views

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

Does the binary operation $m ⋆ n = m^n$ on N have a neutral element? I said yes and it is 1 such that $m ⋆ e = m^e = m$ but apparently that is wrong
2
votes
1answer
22 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
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0answers
31 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
0
votes
0answers
18 views

Conjugacy classes in non-solvable group

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. Also suppose $G$ non-solvable group, $N\unlhd G$, $G/N$ is abelian, $|G/N|=6$ and ...
2
votes
1answer
23 views

Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
1
vote
1answer
30 views

Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
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0answers
28 views

Conjugacy classes in non-abelian simple group

Can we say that every non-abelian simple group has at least 4 non- identity classes?
0
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1answer
32 views

Separating invariant of a group action

Let $G = (\mathbb{R},+)$ be a group, $M = \mathbb{R}^2$, $$\omega \colon \mathbb{R}\times\mathbb{R}^2\to \mathbb{R}^2, \quad \left(t, (x,y)\right) \mapsto(x+t,y-2t)$$ and $$\iota \colon\mathbb{R}^2 ...
1
vote
1answer
29 views

Operations on power set

Which kind of operations on a power set leads to a group/monoid? Known to me are: - intersection - union - symmetric difference - complex product of a group Ate there some more examples?
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0answers
35 views

Is classification Groups G'=G^2=[G,G'] have been done? [on hold]

OK. Suppose that G be a finite group such that the first derivation Of G, i.e., G' equals to commutator G and G' i.e., [G,G'], which we denote with G^2=[G,G']. What can be said about this group?
1
vote
1answer
48 views

Isomorphism classes of abelian groups of certain order

working on a practice question about finite abelian groups and just want to see if I am on the right track: Let $H = <(123)(4567),(8\space 9)(10\space 11),(8\space 11)(9\space 10) > \space ...
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votes
2answers
57 views

Does there exist a group in which this property does not hold? [on hold]

Let $g, h \in G$ is there a group where $(gh)^n \neq g^nh^n$?
5
votes
0answers
55 views

Isomorphism group between $\mathbb R$ and one of it's propre subgroups.

Is there a subgroup $H$ of $(\mathbb R,+)$ such that $H \neq \mathbb R$ and $H \simeq \mathbb R$ as groups?
2
votes
1answer
59 views

Why is the dihedral group closed under composition?!

I've been obsessing over this all day now. I understand associativity, presence of inverse elements and identity, but I don't get why a composition of a reflection with a rotation or other reflexions ...
0
votes
0answers
30 views

Semi direct product

Prove that (i) $GL_n(R)= \coprod_{w\in S_n} UwB$ where $w \in S_n$ is a permutation matrix. and $U$ is a subgroup of $GL_n(R)$ consisting of upper triangular matrices with diagonal entries $1$ and ...
1
vote
2answers
35 views

Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
1
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2answers
83 views

Are all torsion groups finite groups?

Are all torsion groups finite groups? I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?
4
votes
0answers
53 views

How often are Galois groups equal to $S_n$?

Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$. Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$ of ...
0
votes
1answer
35 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
1
vote
2answers
52 views

On infinite groups with unique minimal subgroup

Let $\operatorname{Sub}(G)$ be the lattice of all subgroups of an infinite (abelian) group $G$. If $\operatorname{Sub}(G)\setminus\{\{1\}\}$ has a minimum element which is a cyclic subgroup of prime ...
2
votes
1answer
63 views

Presentation of a group: Show that $\langle a|a^2\rangle =\{1,a\}$.

Example: $\langle a|a^2\rangle=\{1,a\}$. After reading the definition of presentation of a group, I find myself cannot understand the above example given. I don't know which part of the definition I ...
0
votes
1answer
28 views

When can a group be decomposed into a direct product of smaller groups?

Is there any general condition that a group must satisfy in order to be decomposable into a direct sum or product of smaller groups? And what happens if one replaces 'direct product' with semi-direct ...
0
votes
0answers
27 views

Definition of $p$-supersolvable groups

"A group is said to be $p$-supersolvable if it is $p$-solvable and every $p$-chief factor has order $p$." Is this an accurate definition of $p$-supersolvable groups?
1
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1answer
46 views

Solvability of a group

What is the intuition behind the solvable groups? It is defined by composition series. Is there any intuitive way to understand it?
2
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1answer
41 views

On a finite group with unique minimal subgroup

EDIT: Let $G$ be a finite group with a unique minimal subgroup. Then we know that $G$ is a $p$-group. Let $p\neq2.$ Is it true that every subgroup $H\le G$ and every quotient $G/N$ have a unique ...
4
votes
0answers
42 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
1
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1answer
31 views

Sylow subgroups of a non abelian group $G$ with $|G|=21$ and $|G|=39$

I am trying to solve the following exercise: ¿How many Sylow subgroups has a non abelian group $G$ of order $21$ and $39$ respectively. I could do the following: a) $|G|=21=3\cdot 7$. I'll call ...
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votes
2answers
51 views

Definition of “the same”

Given a subgroup H of a group G such that $g^{-1}hg \in H$ for all $g \in G$ and all $h \in H$, I need to show that every left coset gH is the same as the right coset Hg. In the context of this ...
2
votes
1answer
62 views

Classification of subgroup of $\mathbb{Z}^n$.

Fix an integer $n\geq 2$, can we list all subgroups of $\mathbb{Z}^n$?