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

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To find the total number of non isomorphic subgroups of $G$ of order $17$

G be a group of order $17$ .$G$ is cyclic and has two subgroups both trivial , but howto find total number of non isomorphic subgroups of $G$ ? Thanks
2
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2answers
46 views

Labelling the Vertices of Dodecahedron

Dodecahedron has 20 vertices. I want to label them by $1,2,3,4,5$ with the following rule. The five vertices of each face should have different labels. Q. What ...
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2answers
31 views

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$.

Find the number of homomorphisms from $\mathbb Z_5 $to $S_5$. Let $\phi $ be a homomorphism.Then $\dfrac{\mathbb Z_5 }{kerf}\cong Im f$.Now $Im f$ is a subgroup of $S_5$ .Since $kerf $ is a subgroup ...
4
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2answers
84 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
3
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1answer
112 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about ...
0
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1answer
50 views

Why “even number of elements in Group” in this question is given?

I am trying to prove one question about group. "If finite group G has identity e and even number of elements, prove that there is "a" (not equal to "e") such that $a*a=e$." I just don't understand ...
3
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1answer
105 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
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3answers
60 views

is complex number under absolute value a group?

I have just started going over abstract algebra. One of the question is $*$ is defined on $\mathbb C$ such that $a*b=|ab|$ I tried to check three axioms : 1) Associativity 2) identity 3) inverse ...
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2answers
38 views

Prove that $S= \langle(1,2,3,4) \rangle$ has 3 conjugates in $S_4$

Some things I know: $S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$ $(2,4) \in N_G(S)$ Number of conjugates = $[G: N_G(S)]$ This seems like such a easy question but it made me realised that I do ...
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2answers
38 views

Law of Exponents for Abelian Groups

Let $a$ and $b$ be elements of an Abelian group and let $n$ be any positive integer. Show that $(ab)^n = a^nb^n$. Is this also true for non-Abelian groups?
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1answer
17 views

isomorphism complex numbers and $W \times \mathbb{R}$

I have two groups $(W = \{z \in \mathbb{C} \mid |z|=1\}, \cdot)$ and $(\mathbb{R},+)$ and the direct product $(W \times \mathbb{R}, *)$ where $*: (W \times \mathbb{R}) \times W (\times \mathbb{R}) ...
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1answer
27 views

Relationship between proper orthochronous Lorentz group $SO^+(1,3)$ and $SU(2)\times SU(2)$, or their Lie algebras

I have seen sources claim that $SO^+(1,3) \cong SU(2) \times SU(2)$, but have seen others claim that only their Lie algebras are isomorphic. Is it true that $SO^+(1,3) \cong SU(2) \times SU(2)$? If ...
1
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1answer
16 views

group generated by union of positive reals and complex numbers with modulus one

Starting from the group $(\mathbb{C}_0, \cdot)$, we have subgroups $W = \{z \in \mathbb{C} \mid |z|=1\}$ and $\mathbb{R}_0^+$ (the strictly positive reals). My question is what the group, generated by ...
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2answers
99 views

Prove that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$

$\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to ...
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1answer
28 views

Sylow p-subgroups

I need to find the Sylow $p$-subgroups of the alternating group $A_5.$ So I need to find the maximal $p$-subgroups of $A_5.$ First of all, what are the elements of $A_5$? I know they are the even ...
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1answer
29 views

Show that $ker(\phi)=H$

Ler $R$ be a ring and $G$ be a group, $RG$ be the group ring and $H$ be a normal subgroup of $G$. So if |$H$| is invertible in $R$, then setting $e_H=\frac {1} {|H|} \widehat {H}$ where ...
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1answer
37 views

Question on simple subgroup $H$ and a normal subgroup $N$, of $G$

This one is a bit strange to me, mainly the third hypothesis. It goes as follows: Given a group (finite) group $G$, and $N, H \leq G$ such that $N$ is normal in $G$, and $H$ is simple ...
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2answers
38 views

Question regarding homomorphism in groups [closed]

Let $f:G\to H$ a group homomorphism and $\mbox{ker}f$ contains n elements. Prove that $\mbox{Im}f$ has either $n$ or $0$ elements.
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1answer
51 views

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$

If $F$ is a free group then $g^2=h^2$ implies $g=h$ for $h,g\in F$. I've been trying to prove this given the definition of a free group $F$: given group $F$ and subset $X\subseteq F$, $F$ is free ...
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0answers
49 views

Finding the $H$-orbit of $W$ using either Magma or Gap. [closed]

Let $G$ be a finite group and $H$ be a subgroup of $G$. Suppose that $W$ is a subset of $G$. How I could find the $H$-orbit of $W$ by using either Magma or Gap (where $G$ acts upon $W$ by ...
2
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1answer
31 views

Proving the existence of a bijection between $U_{mn}$ and $U_m \times U_n$ where $(m,n)=1$ , there by proving Euler $\phi$ is multiplicative

Without proving before hand that Euler's phi $(\phi)$ function is multiplicative , can we prove that there is a bijection between $U_{mn}$ and $U_m \times U_n$ , for any pair of relatively prime ...
2
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1answer
20 views

G-set without fixed point question

This is a qual exam question: Suppose G is a group of order pq with p < q both prime. Prove that if m ≥ q(p-1), then there exists a G-set A with m elements and without fixed points. Is the fixed ...
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0answers
37 views

How can I work out if a certain group presentation implies a certain relation?

I thought that maybe it would be possible to answer this question using the concept of a group presentation. Let $x_1,x_2,\ldots,x_k$ be k different elements of a group G and $k\geq4$. If we ...
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3answers
55 views

Example where a finite group $G$ of order $n$ has no subgroup of order $m$

Using the Fundamental Theorem of Abelian Groups, one can prove that if $G$ is a finite abelian group of order $n$ such that $m$ is a positive integer that divides $n$, then $G$ contains a subgroup of ...
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2answers
47 views

Galois group, algebraic closure over maximal extension

Let $\overline{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. Let $\alpha \in \overline{\mathbb{Q}}\setminus \mathbb{Q}$ and let $K \subset \overline{\mathbb{Q}}$ be a maximal extension of ...
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3answers
80 views

Prove that the following set is a group

Prove that that the following is or is not a group. (a) The set S = $\mathbb{R}$ \ {0} with operation defined by a * b = 2ab for all a and b in S. (On the right side of the equation, the operations ...
2
votes
1answer
38 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
2
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1answer
43 views

Finite group with unique subgroup of each order.

Let $G$ finite group, and suppose $G$ has unique subgroup of each order (which divides $G$'s order) - Show that $G$ is cyclic. I reduced the problem to sylow subgroups of $G$ (they are all normal), ...
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1answer
49 views

Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. [duplicate]

Let $G$ be a finite group. Prove G is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$. I am stuck with this :( Would appreciate your help.
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6answers
60 views

Ring homomorphism $\mathbb{Z}/8\mathbb{Z} \to \mathbb{Z}/5\mathbb{Z}$ implies $1=0$

Let $f$ be an ring homomorphism from $R_1$ to $R_2$ and define $f^*$ as the homomorphism from the group of units of $R_1$ to the group of units of $R_2$. Suppose $f^*$ is surjective, the question is ...
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0answers
42 views

Show that $<(1,2)(3,4),(1,2,3,4,5)> = A_5$? [duplicate]

I'm revising for my Group Theory exam and saw this in a past year paper question, and I'm not sure enumerating all 60 elements of $A_5$ and showing how I can get them using the given generating set is ...
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0answers
32 views

Symmetric group $S_4$ [duplicate]

Let $G$ be the Symmetric group $S_4.$ Give a representative of each conjugacy class of $G.$ Then calculate the size of each conjugacy class. I have no idea how to do this.
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0answers
21 views

Prove $N\cap Z(P)\ne e$ given a $p$-group $P$ and a normal subgroup $N$. [duplicate]

Let $P$ be a $p$-group and let $N\triangleleft P$. Prove $N\cap Z(P)\ne e$ . Here's what I know so far: $P$, being a $p$-group, is nilpotent and therefore is solvable. That means that it has its ...
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0answers
61 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.
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3answers
35 views

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd , a homomorphism?

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd an injective homomorphism ? I can show that if it is a homomorphism ...
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0answers
15 views

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$?

How can projection operator be expressed in form $\frac{1}{d} (I + \sum_i r_i \lambda_i)$? I was reading a papar and found out that the desity matrix in $d$-dimensional Hilbert Space can be expressed ...
0
votes
1answer
40 views

The order of two subgroups product is not greater than the product of two subgroup orders $|AB|\le |A|\times|B|$

Let $G$ be a group, and let $A,B\le G$ be finite subgroups of $G$. The product $AB$ is defined as following : $AB =\{a\times b | a\in A, b\in B\} $ Show that $|AB|\le |A|\times |B|$. My attempt so ...
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1answer
5 views

Symmetry group of plane quadrilateral

Let $Q$ be a (plane) quadrilateral. Show that the order of its group of symmetries $S(Q)$ is less than or equal to $8$. I tried saying that a quadrilateral cannot have more than 4 rotations, ...
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2answers
44 views

Does there exist infinite group $G$ with subgroups $H,K$ of finite index such that $[G: H \cap K] = [G:H][G:K]$ but $G \ne HK$?

Does there exist infinite group $G$ with subgroups $H,K$ of finite index such that $[G: H \cap K] = [G:H][G:K]$ but $G \ne HK$ ?
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2answers
34 views

If $G$ is a group and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$?

If $G$ is a group and $e \ne x \in G$ and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$ ?
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0answers
43 views

What is the group $G=\mathbb Z_2+\mathbb Z_2$?

I'm trying to figure out what is the group $G=\mathbb Z_2+\mathbb Z_2$ where $\mathbb Z_2 = \{0,1\}$ with the operation addition modulo $2$. I tried to find this group by adding elements from ...
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0answers
45 views

Simple question on the unitary representation.

Let $\pi_1,\pi_2,\pi_3$ are all irreducible, unitary representation of some algebraic group G. Then is it ture that $$Hom_{G}(\pi_1,\pi_2 \otimes \pi_3) \simeq Hom_{G}(\pi_1 \otimes ...
3
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1answer
30 views

My proof that $G(x)\to G / G_x$ is injective

Please could someone check my proof that $\varphi : G(x) \to G/G_x$ is injective? The notation is the following: $G$ is a group acting on a set, $G_x = \{g \in G\mid gx = x \}$ and $G(x) = \{gx ...
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2answers
48 views

Set of conjugates of element cannot be contained in the union of two proper subgroups

Edit: Using the excellent hints provided by @SMM, I was able to solve the problem. See my answer below. I've been thinking off and on about this problem over the last couple of days and would ...
3
votes
1answer
61 views

Given a finite set, how to generate all possible groups defined on it?

Just started learning algebra, the "group" concept looks simple but more thoughts are needed. Given a finite set $S$, say, with $n$ elements, how can we generate all possible groups on $S$? Is there ...
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0answers
57 views

If two groups $G$ and $H$ both have a nonzero homomorphism to all other groups, and have only two idepotent homemorphisms, are they isomorphic?

Consider to Groups $G$ and $H$ such that: The only two endo-hom omorphisms (homomorphisms from a group to itself) that are idempotent (any function $f$ such that $f(f(x))=f(x)$ for all x) are the ...
2
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0answers
46 views

Number of permutations of order m

Is there a closed form for the number of permutations (on n letters) that have order m? If not, is there a tight upper bound?
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1answer
56 views

Proving that a subgroup $|H|=p^k$ is a Sylow subgroup of $|G|=p^km$, $m\nmid p$

I'm attempting to prove Sylow's theorems following the sketch described in the Wikipedia article, but I've run into a little hitch since the theorems are presented in a few slightly different forms in ...
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2answers
53 views

Find the right cosets of $H$ in $G$ simple example

Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$ Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$ Answer given: There are two right cosets, they are disjoint ...
3
votes
1answer
50 views

let $x$ be in finite group $G$ and let order of $x$ is $p$. If $h^{-1}xh = x^{10}$ for a finite group show that $p=3$ [duplicate]

Let $G$ be a finite group, $p$ be the smallest prime divisor of $|G|$ and x $\in$ G an element of order $p$. Suppose $ h \in G $ is such that $h^{-1}xh = x^{10}$. Show that $p = 3$. I cant ...