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

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Homomorphism or not?

Consider the function $$\varphi :\begin{align} \mathbb Z_4 &\to \mathbb Z_4\\z &\mapsto 1\end{align}$$ Why is this not a group homomorphism? On the other hand Why $$\psi :\begin{align} ...
0
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0answers
11 views

amalgamation of locally finite groups

It is well known that in category of groups there are Push-outs so it is possible to realize amalgamation in some kind of most free way. My question is what about category of locally free groups? I ...
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0answers
17 views

Consider group G acting on a set X

Consider group G acting on a set X Give examples of: a)The action that is transitive and faithful My Answer: Group G under addition acting on a set of integers Z b)The action that is transitive ...
3
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2answers
101 views

Why is the set of integers with the operation of addition considered a cyclic group?

The first sentence in the Wikipedia article entitled "Cyclic Groups" states that "In algebra, a cyclic group is a group that is generated by a single element". How is this consistent with addition on ...
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1answer
28 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
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0answers
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Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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1answer
24 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
2
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1answer
35 views

Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
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0answers
21 views

Conjugacy classes of solvable groups [on hold]

If $A$ be a subset of solvable group $G$, let $k_G(A)$ be the number of $G$- conjugacy classes contained in $A$. Also let $N$ be a normal subgroup of $G$ of odd order, $\frac{G}{N}$ be an abelian ...
5
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1answer
45 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is ...
2
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2answers
40 views

If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
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0answers
10 views

The relation of “be characteristic subgroup” is transitive? [duplicate]

If $K$ is a characteristic subgroup of $H$ and $H$ is a characteristic subgroup of $G$, then $K$, is characteristic in G? This is, the relation of "be characteristic" is transitive?
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2answers
21 views

The number of $q$-Sylow subgroups cannot be $p$ for prime $p<q$

Since $q>p $, we cannot have $n_q=p $. Here $n_q $ is the number of $q $ Sylow subgroups. Why is the above statement true? This is a statement from Dummit and Foote.
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0answers
13 views

p-primary component of a group

I have been asked to find the $3$-primary component of the group: $$\mathbb{Z_3}\oplus\mathbb{Z_5}\oplus\mathbb{Z_9}\oplus\mathbb{Z_{153}}$$ Now, I know that we define the $p$-primary component of a ...
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0answers
28 views

Identifying a group

When asked to identify the group $\mathbb{F}_4^{+}$, is my explanation below complete? If not, how can I complete it? The field $\mathbb{F}_4^{+} = \mathbb{F}_2[x]/(x^2+x+1)$ consists of the residues ...
2
votes
1answer
27 views

Show $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$

Let $n \geq 2$ be an integer and $i, j \in \{1, 2, ..., n\} $ be distinct elements. Let $\sigma \in S_n$, Show that $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$ let ...
4
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1answer
45 views

Why is there only one group of order $n$ for some non-primes?

I would like to understand for which integers $n$ is there only one group of order $n$. (up to isomorphism). I understand that if $n$ is prime there is only one group of order $n$. In Sloane's OEIS ...
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1answer
17 views

Minimum possible size of generating set of $(\mathbb{Z}_p)^m$

Is it true that the group $(\mathbb{Z}_p)^m$ cannot be generated by less than $m$ elements?
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1answer
35 views

Square element in a cyclic group

Which elements of a cyclic group are squares (an element $g$ of a group $G$ is a square if $g=h^2$ for some $h \in G$)? Here is my solution; is it correct? Let $G = \{ 1,a,a^2, \ldots , a^n \}$ ...
3
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0answers
35 views

“Breaking the symmetry” in solving algebraic equations

I've heard somewhere a discussion about solving algebraic equations before: When solving a quadratic equation, we are essentially doing the following. Observe that ...
2
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2answers
41 views

Is that true for given group if their divisors are isomorphic, then their quotient group are isomorphic?

I think there will be counter examples in both cases: For given group $G$ and its normal group $H,K$ (1) If $H$ and $K$ are isomorphic then $G/H$ are $G/K$ are isomorphic (2) If $G/H$ are $G/K$ ...
5
votes
2answers
53 views

If $|G|=p^n$, then $p^2 \le |G : G^\prime|$. [on hold]

Prove that, if $G$ be a p-group of order $p^n$, then $p^2 \le |G : G^\prime|$, where $G^\prime$ is the commutator subgroup of $G$ and $n \ge 2$.
2
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1answer
44 views

How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$

I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has: 1 element of order 1 (AKA the identity) 3 elements of order 2 4 elements of order 4 I'm considering the set of all automorphisms on this ...
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0answers
30 views

Subgroups of finite index have finitely many conjugacy classes

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugacy classes. What I think, is ...
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3answers
43 views

Cayley graph is a tree iff group is free

I am looking at this proof of this claim that the cayley graph is a tree iff g is a free group with generating set S. For the direction '$\implies $' I see that they have assumed that there are two ...
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0answers
36 views

how many proper subgroups are there in a trivial group{e}? [on hold]

Is there any proper subgroup of a trivial group of order 1? If yes or no how to prove it?
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1answer
59 views

When is $HK \cong H \times K$?

Suppose $G$ is a group and $H$ and $K$ are subgroups such that $G = HK$ and $H \cap K = \left\{e\right\}$, the identity element of $G$. When can we say that $HK \cong H\times K$? I tried to set up ...
2
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0answers
19 views

The relation of determinants between linear transformation.

I am studying the simplicity of PSL and came across the above statement. I don't understand why $\det L_c =c \det L$? (Given two set of basis of the same vectorspace, $v_1,...v_n$ and $w_1,...w_n$, ...
2
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0answers
34 views

Every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$

In this page, it is claimed that if $K$ is a field then every normal subgroup of $GL_n(K)$ either contains $SL_n(K)$ or is contained in $Z$ (the center, which is the scalar matrices). What is the ...
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0answers
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Subgroups of direct product.

I have problem with finding all subgroups of $\mathbb{Z}_n \times D_m$ and $D_n \times D_m$. First, if $H_1 \leq \mathbb{Z}_n (\text{or} \ D_n)$ and $H_2 \leq D_m$, then $H_1 \times H_2 \leq ...
3
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1answer
51 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
2
votes
2answers
43 views

Is this group homomorphism well-defined?

Let $X = \langle a, b \mid aba^{-1}b^{-1} \rangle$ and $Y = \langle a, b \mid aba^{-1}b \rangle$. I want to define $f : X \rightarrow Y$, such that, $f(a) = a$ and $f(b) = b^2$, however I'm having ...
2
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0answers
25 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
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1answer
61 views

Does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?

I recently was told that $(\Bbb{Z}/n\Bbb{Z})^\times$ is cyclic if $n$ is prime. But then this is impossible if it contains $0$. Hence, does $(\Bbb{Z}/n\Bbb{Z})^\times$ contain $0$?
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2answers
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How to descent to smaller groups “by chopping off a node of the Dynkin diagram”?

I read in section 2 of this paper : "There is a well-defined chain to descent from $E_8$ to smaller groups by chopping off a node of the Dynkin diagram." What exactly is here referring to ...
2
votes
1answer
45 views

If two subsets $S,T\subseteq G$ have sum of cardinalities greater than $|G|$, then $S+T=G$ [duplicate]

Let $S$ and $T$ are two subset of a finite group $(G,+)$ so that $|S|+|T|>|G|$, then Prove that $S+T=G$, where $S+T=\{s+t:s\in S ,t\in T\}$ My effort: It is clear that $S+T\subseteq G$ as ...
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0answers
25 views

Equivalence of right and left cosets of two different subgroups.

Let $A$ and $B$ be two (not necessarily equal) abelian subgroups of $S_5$. If $x$ is an element of $S_5$, under what condition is the following satisfied $$xA = Bx$$ Update: The original question I ...
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1answer
32 views

The Union of Two Normal Subgroups is also a normal subgroup

I know the statement The Union of Two Normal Subgroups is also a normal subgroup is false. Is there a counter example to show this? I can prove that the intersection is normal, but I can't disprove ...
3
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1answer
25 views

List the elements and cosets

In the group $\mathbb{Z}_{24}$, let $H=\langle 4 \rangle$ and $N=\langle6\rangle$ a. list the elements in $HN$ (usually write $H+N$ for these additive groups) and $H\cap N$ So I think $H=\langle4 ...
3
votes
1answer
51 views

Second Isomorphism Theorem

There is one little detail in the proof I would very much like to get your opinion of. Look at where I have circled in red: There it seems that they have used that $\mu_2((hn)N)=h$. But isn't ...
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1answer
30 views

Normal subgroup corresponding to a relation

Suppose I have a free group on $n$ elements, $FX$, quotient-ed by an element (say, $\langle a, b \rangle/aba^{-1}b^{-1}$), how do I compute the normal subgroup $N$ of $FX$, such that $FX/N$ matches ...
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Semidirect product.

I have a problem with representation of this : $(D_n \times D_n) \rtimes \mathbb{Z}_2$, where $\mathbb{Z}_2$ acts on $D_n \times D_n$ by exchanging the two components. $D_n = \langle x, \ y \ | \ x^n ...
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0answers
23 views

A small cancellation group does not contain $\mathbb{Z}^3$

I read somewhere that a small cancellation group (ie. a group admitting a presentation statisfying the small cancellation condition $C'(1/6)$) does not contain $\mathbb{Z}^3$, but without a precise ...
2
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2answers
70 views

Proving that a set is infinite

Consider $G\subset M_2(\mathbb{C})$ where $G =\begin{Bmatrix} \begin{pmatrix} a & 10b\\ b & a \end{pmatrix} & | a,b \in \mathbb{Q},a^2-10b^2=1& \end{Bmatrix} $ . Prove that G is ...
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1answer
40 views

Finite abelian group - product of its elements [on hold]

Is there a finite abelian group G such that the product of the orders of all its elements is 2^2009?
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1answer
59 views

Which group of order 16 is this? [on hold]

$G$ is an abelian group of order $16$ that has elements $a$ and $b$ such that $|a| = |b| = 4$ and $a^2$ does not equal $b^2$. What group is $G$ isomorphic to?
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1answer
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Why is the rank of a group is equivalent to the maximum number of independent U(1) generators?

I read here http://motls.blogspot.de/2012/04/exceptional-lie-groups.html that the rank of group is "the maximum number of independent U(1) generators". In my understanding the rank of a group ...
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0answers
23 views

Spontaneous or not spontaneous symmetry breaking? That is the question.

I have the following system of ODEs: \begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ ...
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1answer
20 views

Mathematical Name for Physical Gauge Symmetries

In physics, when talking about a gauge transformation, we always mean two combined transformations. For example, a $U(1)$ gauge transformation is a combination of $$ \psi \rightarrow e^{ia(x)} \psi ...
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1answer
51 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...