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

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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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A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some $n$? I don't see how this is ...
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31 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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80 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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15 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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30 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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42 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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37 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
23 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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33 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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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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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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23 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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40 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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60 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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1answer
29 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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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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2answers
77 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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1answer
28 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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61 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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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 ...
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50 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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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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Homomorphism from $\mathbb{Z}/n\mathbb{Z}$

Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
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2answers
76 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
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1answer
113 views

A group homomorphism proof with composition

Suppose I have groups $X$, $Y$, and $Z$, and I let $f: X \longrightarrow Y$ and $g: Y \longrightarrow Z$ be group homomorphisms. Now, I want to prove that $g \circ f : X \longrightarrow Z$ is a group ...
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94 views

How to prove that $\zeta*\zeta=\zeta$?

Let $F$ be a non-archimedean local field and $\mathcal{O}_F$ the ring of integers in $F$. Let $G_F=GL_2(F)$. Let $\pi_i$, $i=1,\ldots,n$,be non-equivalent finite dimensional irreducible ...
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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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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 ...
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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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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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1answer
61 views

Non-isomorphic unit groups

Show that the group $U_8$ of units modulo $8$ is not isomorphic to $U_{10}$? This is my answer. Check it for me please. Suppose $U_8$ is isomorphic to $U_{10}$, $3 ∈ U_{10}$ and order of $3$ in ...
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1answer
34 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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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 ...
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22 views

Proof of Conjugate Subgroup Isomorphism

Let $G$ be a group, and let $H$ be a subgroup of $G$. Prove that if $a$ is an element of $G$, then the subset $aHa^{-1} = \{g ∈ G | g = aha^-1 \text{ for some } h \in H\}$ is a subgroup of $G$ that is ...
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1answer
21 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
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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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
132 views

Abstract algebra, prove that $(a^m)^n$ =$ a^{mn}$

Let $a$ be an element of group $G$. For any integers $m,n \in \mathbb{Z}$ ($m,n$ can be positive and negative). Prove that $(a^{m})^{n}=a^{mn}$, then show that $(a^{-1})^{-1} = a$ by using what we ...
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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
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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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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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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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52 views

Abelian Groups, Cardinality of A-A

Let (G,+) be an abelian group and let A be a finite subset of G. Prove that |A-A| = |A| iff A is a coset of a subgroup of G.
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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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~ 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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133 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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159 views

If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.

This is not for homework. (I am a grader for a class.) The case in which $G$ is finite is trivial. (That is, use a corollary to Lagrange's Theorem, and set $[G:H] = \dfrac{|G|}{|H|}$, and similarly ...