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

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Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
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36 views

Is an abelian group the sum of a torsion group with a free group?

Is an abelian group $G$ always isomorphic to the direct sum of its torsion group $T$ with a free group? Alternatively, is the quotient $G/T$ always free? By definition, $T$ consists of all elements ...
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65 views

is the abelianization functor (on groups) full?

By abelianization I mean, for any group $G$, its commutator subgroup is the subgroup $[G,G]$ generated by elements of the form $ghg^{-1}h^{-1}$ for $g,h\in G$. Then the abelianization of $G$ is ...
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34 views

Stabilizer of an element

Can we have any kind of algorithm such that we can find list of all possible stabilizers of an element of a permutation group. I'm asking this question because I want prove that stabilizer of one ...
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Suppose $G$ is a group and $x^3y^3 = y^3x^3 ~ \forall ~x,y \in G.$ Let $H = \{x \in G |~~ |x| $ is relatively prime to 3$\}$.

Suppose $G$ is a group and $x^3y^3 = y^3x^3 ~ \forall ~x,y \in G.$ Let $H = \{x \in G |~~ |x| $ is relatively prime to 3$\}$. Prove that elements of H commute with each other $Attempt$: $x^3y^3 = ...
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70 views

Questions about finitely generated nilpotent groups

Suppose $G$ is a nilpotent group of class $c$, where $c > 1$. This means that if we define the lower central series by $l_1(G) = G$ and $l_k(G) = [G,l_{k-1}(G)]$ for $k \geqslant 2$, then $l_{c+1} ...
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50 views

Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
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68 views

Abstract Algebra. Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$.

Let $\mathit{G} $ be an abelian group. Show that the elements of finite order in $\mathit{G}$ form a subgroup of $\mathit{G}$, called the torsion subgroup of $\mathit{G}$. let $g \in G$ I know that ...
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52 views

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field?

Does every finite cyclic group appear as a subgroup of the multiplicative group of a finite field? In other words, given any $d \in \mathbb{N}$, can we find a prime $p$ and $k \in \mathbb{N}$ such ...
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18 views

Clarification on a cycle parity proof

Prove that cycle $(a_1a_2...a_k)$ is even $\iff$ $k$ is odd. This makes intuitive sense because $(a_1a_2a_3...a_k)=(a_1a_k)(a_1a_{k-1})...(a_1a_3)(a_1a_2)$ which will be an even number of ...
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79 views

If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian

I try to prove the following: Let $G$ be a group and $N$ be a normal subgroup. Suppose $N$ is abelian and $G/N$ is abelian. Then $G = N \cdot (G/N)$ is abelian. Proof: For the first assertion ...
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73 views

Non-commutative quotient group?

If you have a non-abelian group $G$ with some normal subgroup $K$, is it possible to have a non-abelian quotient group $G/K$? Besides actually sitting down and trying to generate quotient groups ...
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19 views

Number of different roulettes

Problem: A casino roulette is divided in $n$ sectors: each one is coloured by a chosen colour between $p$ distinct colours. What is the number of distinct possible casino roulettes? (we don't ...
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48 views

Isomorphism between this subgroup of complex numbers and all finitely generated abelian groups ?

Every cyclic (abelian) group of infinite order is isomorphic to $G=(\mathbb{Z},+)$. Is there a corresponding set of groups $S_G=\{G\}$ such that every finitely generated abelian group of infinite ...
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98 views

Abstract algebra: Proof that if $\mathit{H}$ is a subgroup of index 2 in a finite group G, then gH=Hg for all g $\in$ G

Proof that if $\mathit{H}$ is a subgroup of index 2 in a finite group G, then g$\mathit{H}$=$\mathit{H}$g for all g $\in$ $\mathit{G}$. I do know that index, or $\frac{|G|}{|H|}=2$ implies that ...
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36 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
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46 views

Finite group $G$ is product of a subgroup $H$ and normalizer of a Sylow $p$-subgroup of $H$

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of $H$. Let $N_G(P)$ be the normalizer of $P$ in $G$. Show that $G=N_G(P)H$.
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60 views

Sylow subgroups of soluble groups

Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
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171 views

Classifying abelian groups up to isomorphism

List all abelian groups (up to isomorphism) of the given orders: a) $144$, b) $600$ a) For order $144$, I feel confident with this one so far: $\mathbb{Z}_4 \oplus \mathbb{Z}_{36}$ Elementary ...
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93 views

Group categories with only one object with a defined product

Do you know how to deal with this kind of problem? Let $G$ be a group and $\mathcal {G} $ be the category with one object $G$ and $\mbox {Mor}( {G} ; {G} ) = {G} $. Find all groups $G$ such that ...
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What is $\mbox{Aut}\left(\mathbb{Z}_{n}\right)$?

I understand that $\mbox{Aut}\left(\mathbb{Z}_{n}\right)$ is the group of all automorphisms under function composition, but I am a little confused about the sort of group it forms. If the elements of ...
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44 views

Group Scheme - equivalent ways of defining it

I know that a group scheme is an $S$-scheme $G$ equipped with one of the equivalent sets of data a triple of morphisms $μ$: $G$ ×S $G$ → $G$, $e$: $S$ → $G$, and $ι$: $G$ → $G$, satisfying the ...
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57 views

what is degree of permutation group?

Is "degree" the same term as "order" of a permutation group?
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66 views

How many normal subgroups of the free group on $n$ generators?

Let $F_n$ be a free group on $n>1$ elements and $\mathcal{N}(F_n)$ the set of normal subgroups of $F_n$. Since there are uncountably many nonisomorphic groups on two generators we have that ...
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73 views

Prove the order of an element is the order of the group

If $g \in G$ and $|G| = n$, prove $g^n = e$. I actually don't know if this is true or not, which is why I am trying to prove this. I know that the order of $g$ is $n$, but I don't really understand ...
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92 views

Embedding $S_n$ into $A_{n+2}$

I am trying to prove that for all $n$, $S_n$ is isomorphic to a subgroup of $A_{n+2}$. Say $S_n$ acts on $\{\alpha_1,...,\alpha_n\}$ and $A_{n+2}$ acts on ...
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38 views

Characterizing $\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q})$ for $p_i$ primes?

For what $n$ is $$ \operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q}) $$ known, where the $p_i$ are primes? By Kummer theory, I think that $$ ...
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102 views

Determine whether or not G is isomorphic to H.

$G$ is $S_3$. $H = \{I,A,B,C,D,K\}$ where $$ I= \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} \quad A=\begin{pmatrix} 0 & 1 \\ 1 & ...
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75 views

$\mathbb{Z}G$ is necessarily a subgroup of $G$, or am I missing something?

Let $\mathbb{Z}G=\left\{\sum\limits_{g\in G}n_gg\mid n_g\in\mathbb{Z},g\in G\right\}$, where if $G$ is infinite, we only consider finite formal sums of elements of $G$ with coefficients in ...
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1answer
41 views

Why is $H_1 \le G \land H_2 \le G$ necessary in $a(H_1 \cap H_2) = aH_1 \cap aH_2$?

The problem is as follows: $G$ is a group and $H_1$ and $H_2$ are its two subgroups (i.e., $H_1 \le G \land H_2 \le G$). To prove that $a(H_1 \cap H_2) = aH_1 \cap aH_2$. Here is my trial: On ...
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Counting Groups

My task is to determine the number of Abelian groups of order $p_{1}^{4}p_{2}^{4}...p_{n}^{4}$, where each $p_{n}$ is a distinct prime. My attempt: $\forall p$, there are 5 possible non isomorphic ...
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54 views

Find all homomorphisms from $D_4$ to $\mathbb{Z}_4$

My professor put this question on a practice test with a hint that there is more than one but less than five. $$D_4=\{I,R,R^2,R^3,F,RF,R^2F,R^3F\}$$ I started looking at the order of the elements ...
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49 views

Grothendieck Group for General Monoids

Is there an analog to Grothendieck group for general monoids? I imagine such thing could be constructed but I have not found this construction in standard text so I am not sure if it makes sense. (Or ...
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29 views

Can anything be said for the topology of a topological monoid?

A topological group is one in which the group operations (the multiplication and inverse) are continuous, or equivalently as a group object in $\mathbf{Top}$. They are uniformisable and hence are ...
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103 views

Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
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69 views

Explain why there is only one homomorphism from D6 to C5

I have two questions that I'm struggling with: a) Give an example of a quotient group G/N, where G is not commutative (non abelian) but G/N is commutative (abelian) EDIT: I can find a specific ...
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33 views

Relationship between normal subgroups and intersections

Let $G$ be a group, $N_1,N_2\leq G$ normal subgroups of $G$ and $H_1,K_1\leq G$ normal subgroups of $G$ such that $N_1\cap N_2\subseteq K_1\subseteq H_1$. As the quotients $H_1/N_i$ and $K_1/N_i$ ...
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73 views

Is this relaxed definition of a group true?

Let $G$ be a non-empty set and let $*$ be a binary algebraic operation. Algebraic system $<G, *>$ is a group if $\forall a \in G, \forall b \in G, \forall c \in G: (a * b) * c = a * (b * c)$ ...
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30 views

What is PSU(1,1) PSU(2,1)? Is there a general group of the form PSU(n,m)?

I have seen it written about groups PSU(1,1), PSU(2,1). But what exactly are these? The definitions were not given, and I can't seem to find a definition online. Moreover is there a general class of ...
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26 views

Subgroups of PSL(2,R): criteria for discreteness

Let $G$ be a subgroup of $SL_2(\mathbb R)$ generated by a set of matrices $\mathcal M=(M_i)_{i\in I}$. Is there an effective criterion on $\mathcal M$ ensuring that $G$ is discrete?
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97 views

Generating the symmetric group $S_n$

I know that $\sigma =(1 2 \ldots n)$ and $\tau =(1 2)$ together should generate the symmetric group by virtue of conjugation, i.e. $(\sigma)^k \circ \tau \circ (\sigma^{-1})^k = (k+1, k+2)$; we know ...
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25 views

Subgroups, orders and divisibility.

Show that for any two subgroups $H$ and $J$ of $\Bbb Z_n$, we have $H ⊆ J$ iff $|H| \text{ divides }|J|$. Attempt: Suppose $H ⊆ J$, then we know there are generators that generate $H$ and $J$ since ...
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39 views

How to prove its maximality of normal core?

For a group $G$, the normal core $H_G$ of a subgroup $H \le G$ is defined as the intersection of the conjugates of $H$, i.e., $$H_{G} = \bigcap_{a \in G} a H a^{-1}.$$ It is remarked that $H_{G}$ is ...
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45 views

About some special kinds of group automorphisms

let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$. Is there any non-trivial ...
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Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
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Isomorphism Classes

I'm trying to find what the isomorphism class of the group of $G = n\times n$ matrices with $\pm 1$ along the diagonal and zeroes everywhere else. My approach was first to show that because for each ...
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61 views

If $G$ is non-abelian group of order 6, it is isomorphic to $S_3$

Let $G$ be a non-abelian group of order $6$ with exactly three elements of order $2$. Show that the conjugation action on the set of elements of order $2$ induces an isomorphism. I just need to show ...
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60 views

Group action on coset space. prove ker of associated homomorphism is largest normal subgroup

Prove that left multiplication defines an action of $G$ on the coset space $\frac{G}{H}$ and that the kernel of the associated homomorphism from $G$ to $\operatorname{Aut}\left(\frac{G}{H}\right)$ is ...
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46 views

Question over group isomorphisms

Let $G$ and $H$ be two groups, and denote the direct product, $G \times H$, by K. The direct product is defined as: Let $(S, \bigodot$) and $(B, \bullet)$ be groups with the respective operation. ...
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78 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...