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

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Can a quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
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0answers
11 views

Roots and Weights

I use a Mathematica package to compute roots and weights (and other things) but the package gives me only the expression of the roots in $\omega$-basis (basis of fundamental weights) and in the ...
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1answer
28 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
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0answers
16 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
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3answers
26 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
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1answer
31 views

Finding Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$. The mistake in this method?

We need to find the Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$ . This method does not give me the right answer (i.e $6$ ) . Attempt: We need to find the number of Cyclic ...
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1answer
22 views

Automorphisms of group extensions

Assume we have a group extension $1 \to N \to G \to H \to 1$, and an automorphism $\phi: G \to G$. Is it correct that this automorphism induces automorphisms $\phi_N : N \to N$ and $\phi_H : H \to H$ ...
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1answer
54 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$ where $|a|=m_0$ and $|b| = m$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some ...
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4answers
94 views

Do we have $(G/H)\times H \cong G$ for groups in general?

After some thought I began to suspect $(G/H)\times H \cong G$, so I tried to construct an isomorphism by hand. I came up with $\varphi: (gH, h) \mapsto gh$ which came out to work provided $G$ is ...
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1answer
19 views

the smallest quasigroup, which is not a group

I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?
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1answer
44 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
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1answer
55 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
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1answer
26 views

Range and kernel of groups

Let $f: G \rightarrow H$ be a homomorphism. If the range of $f$ has $n$ elements, then $x^n \in$ ker $f$ for every $x \in G$. I can kind of understand why this is true. The ker of $f$ is $\{x \in ...
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1answer
37 views

Cyclic and abelian group

A group $G$ has order $25\cdot 47\cdot 17$. Is it cyclic and/or abelian? I know that a group of order $47$ or $17$ is cyclic, should I somehow use it?
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1answer
21 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
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2answers
40 views

Non-abelian group of order 28 which is not the dihedral group

Consider the group of order 28 with Sylow $2$-subgroups that are cyclic. We can derive that the Sylow $7$-subgroup is normal, and that this group is uniquely determined by the relation $bab^{-1}=a^6$ ...
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1answer
70 views

If $G$ is solvable and $G/[G,G]$ is cyclic, can $G\times G$ be generated by 2 elements?

I doubt this is true, but I haven't found any small counterexamples (there are no counterexamples with $|G| < 1536, |G| \neq 768$): Suppose $G$ is finite, solvable, 2-generated, and $G/[G,G]$ ...
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1answer
24 views

A group generated by two elements such that its product with itself is not generated by two elements.

We have $S_5=\langle (12345), (12)\rangle$ and we can show that $S_5\times S_5$ is also generated by two elements. Is there a group $G$ generated by two elements such that $G\times G$ is not generated ...
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0answers
30 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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0answers
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Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
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0answers
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Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. Condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle ?$

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. State a necessary and sufficient condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle$ Attempt : Let $l = ...
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1answer
31 views

$[G:\cap H_i]\leq\Pi[H_i:H_{i+1}]$

If $H_0=G$ and $H_{n+1}\subseteq H_n\subseteq G$ for $n\in \mathbb N$, then $[G:\cap H_i]\leq\Pi[H_i:H_{i+1}]$. I used Poincare inequality, but it doesn't work.
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2answers
44 views

Inn characteristic in Aut

If $G$ is a centerless group then is $\mathrm{Inn}(G)$ necessarily characteristic in $\mathrm{Aut}(G)$? The condition of being centerless is necessary as $D_8$ provides a counterexample otherwise.
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1answer
62 views

Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
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0answers
18 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
3
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2answers
65 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
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4answers
216 views

How is this subgroup normal?

Let $G$ be a group, and let $U$ be a subset of $G$. Let $\hat{U}$ be the smallest subgroup of $G$ containing $U$. Then $\hat{U}$ is the intersection of the collection of all the subgroups of $G$ ...
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0answers
21 views

Groups and Homomorphism [on hold]

let $f:G\to G'$ be a homomorphism. Prove that if $H'$ is a subgroup of $G'$ then $f(H')=${$x|f(x) \in H'$} is subgroup of G. if $H'$ is a normal subgroup of $G'$ then $f(H')=${$x|f(x) \in H'$} is ...
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1answer
31 views

Surjective Homomorphism Symmetric group

For $G=S_4$ i'm having a bit of trouble following the solution. For the blue underline I was wondering if there is a strategy for spotting this relatively quickly. For the green underline I ...
3
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0answers
38 views

If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...
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2answers
22 views

Requested material on Bilinear Pairing

Bilinear map/pairing is widely used in Pairing based Cryptography. I am new to this area. Can anyone suggest me some good reference on Bilinear pairings? I need at least an example of Bilinear map ...
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0answers
18 views

determining maximum number of elements of particular order

In direct product of an infinite group, say of nonzero reals or positive reals, is there a way to determine a number of elements of a particular order (e.g., 1 or 2), or at least know whether the ...
2
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1answer
29 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
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1answer
22 views

order of an element formula

I was wondering whether there's a formula or something. If it is given that $x^n = e$ and $x^m = e$, does it mean $x^{gcd(n,m)} = e$, so we can determine whether $x=e$ or $x \ne e$?
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4answers
93 views

proof a function is an isomorphism

When we prove a function is an isomorphism, we need to prove it's a bijection and it's closed under an operation. In one example I had no problem proving the first part, but in the second part, I ...
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1answer
35 views

How do I show a mapping is a homomorphism?

I don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and more ...
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2answers
56 views

Confusion related to definitions involving free groups

From Wikipedia ...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different ...
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1answer
18 views

order of elements in direct product

I have a conceptual question: if a group has 1 element of order 1 and 1 element of order 2 (e.g., nonzero reals), what changes if your take its direct/cartesian product as a group?
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2answers
348 views

Is the Axiom of Choice implicitly used when defining a binary operation on a quotient object?

Let's say you have a group $(G,\cdot)$ and you have a normal subgroup $N$ (note we are considering this only as a set). And now we want to define a binary operation $\star$ on $G/N$ such that $(G/N, ...
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2answers
40 views

Subgroup of size p p=prime

Are subgroups of size $p$ where $p$ is prime, cyclic subgroups? I understand that if the group $G$ is prime order, then $G$ is cyclic.
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1answer
29 views

Surjective Homomorphism Dihedral Groups [on hold]

For the first underlining i'm just wondering where i have used the fact that $\phi$ is surjective. For the second underling I don't understand the explanation it gives for why $<p^2>$ is the ...
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2answers
78 views

Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups?

Well, this is my question. Is every subgroup of the product of two cyclic groups is again a product of two cyclic groups (maybe one being trivial)? Thanks!
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3answers
63 views

Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am lookking for ...
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1answer
24 views

Lang's proof of Cauchy's Theorem

In proving Cauchy's theorem in his 'Algebra', Lang first prove[s] by induction that if $G$ has exponent $n$ then the order of $G$ divides some power of $n$. Let $b \in G, b \ne 1$, and let $H$ be ...
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1answer
36 views

Necessary and sufficient for $\operatorname{orb}(x)=\operatorname{orb}(y) \iff \operatorname{Stab}(x)=g\operatorname{Stab}(y)g^{-1}$

Are orbits equal if and only if stabilizers are conjugate? You may get some insights from the link above. My Question: What is the necessary and sufficient condition for the above statement to be ...
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0answers
17 views

Meaning behind the conjugacy class in describing geometry of solids

When we consider the group of rotataional symmetry , say of a cube or a dodecahedron, it is not difficult to see the symmetry group is isomorphic to a $S_4$ , $A_5$ respectively. Mooreover, when you ...
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0answers
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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3answers
134 views

Can we find an element of infinite order in a symmetric group of infinite order?

In particular, I'm thinking of a simple example: the group $S_\Omega$ given $\Omega = \{1, 2, 3, ...\}$. I've been thinking of elements of $S_\Omega$ in terms of their cycle decomposition, which may ...
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0answers
24 views

How do you find a minimum of a function with these tools?

Let's say I can define a group $G$ acting on a set of combinatorial objects $X$ and I have a function $f: X \to \Bbb{N}$ that I want to find a minimum of in $X$. Is there a polynomial time ...
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2answers
41 views

If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$

Let $(G,\cdot)$ be a group and let $\mathfrak{S}(G)$ be the set of all bijective mappings from $G$ to $G$. Show that: If $G$ is non-abelian, then $Inn(G):=\{\kappa_a \vert a\in G\}$ is not a ...