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

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On the Frattini Subgroup

For a prime $p$, let $H=\{x\in \mathbb{C}\colon x^{p^n}=1 \mbox{ for some } n\geq 1\}$ be the Prufer $p$-group, $C_2=\langle y\colon y^2=1\rangle$, and $G=H\oplus C_2$. Then $H$ is the unique maximal ...
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94 views

Given a group homomorphism $f:G\to H$, if $m$ is relatively prime to $|H|$ and $x^m\in\ker f$, then $x\in \ker f$

Let $f:G\to H$ be a homomorphism, and let $m$ be an integer such that $m$ and $|H|$ are relatively prime. For any $x \in G$, if $x^m \in \ker f$, then $x \in \ker f$. My proof step: if $x^m \in ...
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150 views

Hall $\pi$-subgroup and $HN=G$

Let $\pi$ be any set of prime numbers. A finite group $H$ is a $\pi$-group if all primes that divide $|H|$ lie in $\pi$. If $|G|<\infty$, then a Hall $\pi$-subgroup of $G$ is a $\pi$-subgroup ...
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230 views

Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible ...
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143 views

Determining the group homomorphism in semidirect product

We know that if $N$ is a normal subgroup, $H$ is a subgroup, and $\varphi$ is the group homomorphism such that $\varphi:H\to$Aut$(N)$. And this gives a unique group, called the outer semidirect ...
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48 views

Simple meaning to Center of a group

Recently I was learning Center of groups and on referencing the group table, I observed is that all the rows that are also present as columns are the centers of any group. So, I made a small program ...
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127 views

Explain the formula for the size of an orbit…

Explain the formula for the size of an orbit and show that this always is a divisor of the group order $|G|$. (You may use Lagrange's theorem!) So I would like to know how i can go about answering ...
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103 views

Sylow $p$-subgroups in Infinite Groups

I was thinking about the following problem and I hope some of you can answer me. Let $G$ be a nilpotent group. Let $P$ be the $p$-Sylow subgroup of $G$. What about the $p$-subgroup of an homomorphic ...
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49 views

Basic group theory question on countable groups and infinite abelian groups.

Here are two (maybe simple) questions. A: Every countable group $G$ has only countably many distinct subgroups. B: Every infinite abelian group has atleast one element of infinite order. Both these ...
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63 views

normal subgroups: proving $H\lhd D_8$, given $K \ntriangleleft D_8$, knowing $KH$

I was asked to prove $H\lhd$$D_8$, given that K$\ntriangleleft$$D_8$. But later in the same problem I am suppose to prove that $HK\lhd$$D_8$. I thought that if $H\lhd$$D_8$ and $K\lhd$$D_8$ then ...
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27 views

Operation of $O_2$ on the plane.

I am currently reading through M. Artin's Algebra 2.ed. In chapter 6, remarks similar to the following are often made with little explanation: "Unless an origin is chosen, the orthogonal group $O_2$ ...
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92 views

Is G isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$?

If $ G=\{3^{m}6^{n}|m,n \in \mathbb{Z}\}$ under multiplication then i want prove that this G is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$.Can any one help me to solve this example? please help me. ...
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454 views

In how many different ways can the faces of a regular dodecahedron be colored?

prove that there are 9099 different ways of colouring the faces of a dodecahedron red, white or blue. (this is from Amstrong's "group and symmetry") attempt: Since the question refers to the faces of ...
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160 views

Showing that a specific function $\mathbb Z[x]\to\mathbb Z$ is a group homomorphism.

Let $\mathbb Z[x]$ be the group of polynomials in an indeterminate $x$ with integer coefficients under addition. Prove that mapping from $\mathbb Z[x]$ into the group $\mathbb Z$ given by mapping ...
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86 views

How to find the order of these groups?

I don't know why but I just cannot see how to find the orders of these groups: $YXY^{-1}=X^2$ $YXY^{-1}=X^4$ $YXY^{-1}=X^3$ With the property that $X^5 = 1$ and $Y^4 =1$ How would I go about ...
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222 views

Show that r is a primitive root?

Show that if $r$ is a primitive root modulo the positive integer $m$, then $ {\overline r }$ is also a primitive root modulo m if $ {\overline r }$ is an inverse of $r$ modulo $m$. My TA did not go ...
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48 views

$HK\cap N=H(K\cap N)$

I'm trying to prove If $H$, $K$ and $N$ are subgroups of a group $G$ such that $H\lt N$, then $HK\cap N=H(K\cap N).$ I'm trying sets inclusion to prove it, am I in the right way? I need help. Thanks ...
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56 views

Studying the action of $GL(V)$ on the vector space $V$

The statement I am trying to prove is the following. Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let $\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural ...
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85 views

Inner Automorphisms with Groups

Let $G$ be a group, and let $g \in G$ . Prove that the function $\gamma_g: G \to G$ defined by $(\forall a \epsilon g): \gamma_g(a)=g a^{-1} g $ is an automorphism of G. The automorphisms $\gamma_g$ ...
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28 views

Groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.

I cant solve this exercise. Find all groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$. I need a little help here. thanks!!!
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53 views

Choosing central generator for nilpotent group generated by 3 elements

Let $G$ be a group, which is 2-step nilpotent torsion-free generated by three elements (in a minimal presentation) such that the centre of $G$ is generated by one element (i.e. $C(G)$ is infinite ...
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101 views

Groups of units: Find an explicit isomorphism $U_{35}$, $U_{39}$

I need help in the following exercise: Find an explicit isomorphism between $U(\mathbb{Z}/35\mathbb{Z})$ and $U(\mathbb{Z}/39\mathbb{Z})$. Thanks!
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93 views

Isomorphisms of the Lorentz group and algebra

I'm trying to read a few books on QFT and some seem to say the Lorentz algebra obeys $\mathfrak{so}(1,3)\otimes \mathbb{C} \cong \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ while others say ...
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80 views

Cyclic group of direct product

I know that $Z_m \times Z_n$ is cyclic and isomorphic to $Z_{mn}$ if and only if $\gcd(m,n)=1$. There is also a corollary that saying "The group $Z_{m_1} \times Z_{m_2} \times Z_{m_3} \times \ldots ...
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35 views

Let $a_i$ and $a_j$ disjoint cycles of $S_n$. Since $τ∈S_n$ is an one-to-one function, $τα_i τ^{-1}$ and $τα_jτ^{-1}$ must be also disjoint.

My solution manual says that: Let $a_i$ and $a_j$ disjoint cycles of $S_n$. Since $τ∈S_n$ is an one-to-one function, $τα_i τ^{-1}$ and $τα_jτ^{-1}$ must be also disjoint. This make sense to ...
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88 views

Euler's formula and subgroups of $\mathbb Z_n$

Prove that in $\mathbb Z/n\mathbb Z$ for every divisor $d$ of $n$ there is a unique subgroup of order $d$ using the following results: $\sum_{d\mid n}\varphi(d)=n$ and the number of generators of ...
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45 views

Property of subgroups

Is the following property correct? Property: let $(G;f_G)$ group, $B \subseteq G$ and $B \neq \emptyset $, then $(B;f_G|_B)$ is group iff $\forall a,b \in B$ we have that $a f_G|_B b'\in B $ and $ b ...
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130 views

If $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$ , then $A\cap B$ is a normal subgroup of $B$

Suppose $A$ is a normal subgroup of $G$ and $B$ is a subgroup of $G$. Show that $A\cap B$ is a normal subgroup of $B$.
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139 views

Sp(1) and SO(3) Locally Isomorphic - Trouble with part of proof

My apologies if this is fairly basic. I'm trying to understand the proof that $Sp(1)$ and $SO(3)$ are locally isomorphic but are not isomorphic but have run into some trouble. Here's what my ...
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165 views

An alternative definition of a group?

Will the following definition of a group work as a basis for group theory: $\forall G,f,i,e (Group(G,f,i,e)\leftrightarrow f:G\times G\rightarrow G$ $\wedge i:G\rightarrow G$ $\wedge \forall ...
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91 views

Sylow subgroup of the normalizer of itself

So I was given a finite group $G$, with $P \leq G $, P is a p subgroup of $G$, and $P\in Syl_p(N_G(P))$. I want to show that P is a Sylow p subgp of $G$. So I attempted a contradiction, supposing ...
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278 views

Normal Subgroups and Quotient Groups

Let $G$ $(x^i y^j)$, $i = 0,1$, $j =0,1, 2, 3, \ldots, n-1$, where $(x^i y^j) = (x^{i'}y^{j'})$ iff $i = i'$, $j = j'$ $x^2 = y^n = e$, $n>2$ $xy = y^{-1} x$. a) Find the form of the product $(x^i ...
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58 views

How I can doing a sum of a vector in $ℤ^{r}$ with a equivalence class in $ℤ/nℤ$

Let us consider the following direct sum of groups: $G=ℤ^{r}⊕ℤ/nℤ$ My question is: I know that every $w$ in $G$ can be written as: $w=u+v$ where $u$ is in $ℤ^{r}$ and $v$ is in $ℤ/nℤ$. However, I am ...
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30 views

quotient group inheritance of some properties from original group

Let us say that there is the group with addition modulo 6: $G = \{0,1,2,3,4,5\}$ and let $N = \{0,3 \}$. Then the quotient group $G/N$ would be $G/N = \{ \{0,3\}, \{1,4 \}, \{2,5\} \}$. According ...
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37 views

Enumerate Elements in Abelian Group

So I am reading in my book, and some across this example: Consider the group $\mathbb{Z}^*_{15}$. We can enumerate its elements as: $[\pm 1], [\pm 2], [\pm 4], [\pm 7]$ Can someone explain how the ...
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345 views

Does every nontrivial finite cyclic group have prime order?

Here's my attempt as a proof. Did I make any mistakes? Let $G \ne \langle e \rangle$ be a finite cyclic group. Then any $a \in G \setminus \{e\}$ has order $|G|$. Aiming for a contradiction, ...
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59 views

Prove elements of $D_8$ and $\left \langle (1\; 2\; 3) \right \rangle$ do not commute in $S_4$

The complete question is: Fix any labelling of the vertices of a square and use this to identify $D_8$ as a subgroup of $S_4$, then prove that elements of $D_8$ and $\left \langle (1 \;2 \;3) \right ...
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88 views

Normal Subgroups, index, divisible orders

Let $H$ be a normal subgroup of $G$ with index $k$ . Show that if $a \in G$ and $o(a)=n$, then the order of $aH$ in $G/H$ divides both $n$ and $k$ .
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Proving $f(A)\leq Z(H)$

Please help me to prove that if $f: G\to H$ is surjection and $A\leq Z(G)$ then $f(A)\leq Z(H)$. Thank you.
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61 views

Direct sum of a group and generator

When talking of a group $G$ being the direct sum of a set of subgroups $\{H_i\}$, the following part is a part of definition (from wiki): $G$ is generated by the subgroups $\{H_i\}$ So, does this ...
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204 views

Let $f:G→G'$ a homomorphism and let $K=\ker(f)$ and let $H$ an subgroup of $G$. Prove that: $f^{-1}(f(H))=HK=\{hk:h∈H,k∈K\}$

Let $f:G→G'$ a homomorphism and let $K=\ker(f)$ and let $H$ an subgroup of $G$. Prove that: $$f^{-1}(f(H))=HK=\{hk:h∈H,k∈K\}$$
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45 views

$s_{l(α)}=r_{α/2}\circ s_x$

Let $l(α)$ be a line through the origin, such that the angle between $l(α)$ and the $x$-axis is $α$. Let $s_{l(α)}$ be a reflection in the line $l(α)$. And let $s_x$ be a reflectioin in the $x$-axis. ...
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113 views

Very poor proof of order of elements in a permutation group

In the group $S_{10}$ there is an element of order $30$. Also prove that no element has order $11$. (i) $(1,2,3,4,5)(6,7,8)(9,10) = 5*3*2= 30$ How do they view this as an element i wanna look ...
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107 views

Group theory - simple subset proof

Let $G_{1}, G_{2}, G_{3}, G_{4}$ be four finite subgroups of the group $G$. Is $(G_{1}\cap G_{3}) \circ (G_{1}\cap G_{4}) \subseteq G_{1} \cap (G_{3}\circ G_{4})$ true? My attempt: Let $x \in ...
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87 views

Calculate $\operatorname{Hom}_{\Bbb Z}(\Bbb Z_6,\Bbb R^*\oplus \Bbb C^*)$ [duplicate]

Let $\Bbb R^*=\Bbb R-\{0\}$ (non-zero real numbers) and $\Bbb C^*=\Bbb C-\{0\}$ (non-zero complex numbers) be multiplicative groups. Is this equality true? $$\operatorname{Hom}_{\Bbb Z}(\Bbb ...
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76 views

Necessary Condition For Having A Normal Subgroup Of A Non Abelian Group

Is there any necessary condition for non abelian group to have a normal subgroup?
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70 views

Showing that a subgroup $H\subseteq G$ is contained in its normalizer $N_G(H)$

Suppose $G$ is a group and $H \leq G.$ Define a set $$N_G(H)=\lbrace a \in G:aHa^{-1}=H \rbrace.$$ Show that $H \leq N_G(H)$. I try to use the statement '$H \leq G \iff ab^{-1} \in H\,\,\forall ...
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124 views

Degree of a permutation group

What can we say about the set that a group can act on it as a permutation group? we know that this set is not unique. For example the alternative group $A_4$ acts on the sets of sizes 4 and 6.
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56 views

group presentations that are the most symmetric

for a group G, does there exist a group presentation which is the most "symmetric" by which I mean has the most automorphisms of the group by permuting generators?
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358 views

$ U(1) $ and $ SO(2) $ are locally equivalent.

In one of my particle physics textbooks, I came across the statement above. I don’t really know what it means. I know a bit of group theory and that $ U(1) $ is just the $ 1 $-d unitary transformation ...