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

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7 views

Prove that $SO_2(\mathbb {C})$ is unbounded.

Can someone help proving that $SO_2(\mathbb {C})$ is unbounded.
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1answer
20 views

Exponents of the Coxeter group A(n)

I came across the following result : "The exponents of the Coxeter group A(n) (n>=1) are 1, 2, ... , n." I am not able to figure out a proof of this fact. Any help towards proving this result will be ...
2
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2answers
31 views

$G$ contains at least $r(p-1)$ elements of order $p$

Suppose a group $G$ has $r$ distinct subgroups of prime order $p$. Show that $G$ contains at least $r(p-1)$ elements of order $p$. Aside: I know how to use this to prove that a group of order $56$ ...
3
votes
1answer
30 views

HK is cyclic if H and K are both cyclic

Let $G$ be a finite group with normal subgroups $H$ and $K$ of relatively prime orders. Show that the group $HK$ is cyclic if $H$ and $K$ are both cyclic. My attempt was to use the $2$nd Isomorphism ...
3
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2answers
59 views

Shouldn't this be “injective” rather than “well-defined”?

Consider the top of page 6 here: "...since different motions might place the globe in the same position think about why this group operation is well-defined..." He is talking about the group of ...
6
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1answer
82 views

Group of Order 33 is Always Cyclic

I would love to get help on this problem from a chapter on Commutator of Group Theory: Show that each group of order 33 is cyclic. (Hint: Use the result from the Exercise and Lemma below.) ...
3
votes
1answer
43 views

A question about a free abelian finitely generated group.

I am having a hard time solving this and it is really confusing. I don't have enough schema, which makes it problematic. Let $A$ be a finitely generated free abelian group and $B$ is a subgroup of ...
1
vote
1answer
33 views

Finding suitable basis for a free abelian finitely generated group.

I am stuck with this exercise forever... I was barely taught about it, English is not my mother language and in any other phrasing it is not coherent with my material.I'd really appreciate your help. ...
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1answer
48 views

Prove or disprove $a^{\varphi(m)}=a^{k\,\varphi(m)}\pmod{m}$

Prove or disprove that $a^{\varphi(m)}=a^{k\,\varphi(m)}\pmod m$ where $\varphi(m)$ is the totient function, and $k\ge1$. This is clearly true when $\gcd(a,m)=1$ since both sides are $1$ due to ...
1
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0answers
28 views

$\mathrm{Aut}(G)$ vs. $\mathrm{Aut}(H)$ where $H$ is a maximal abelian subgroup

Can I find a finite group $G$ and a maximal abelian subgroup $H$ such that $ \mathrm{Aut} (H)$ is not isomorphic to a subgroup of $\mathrm{Aut}(G)$?
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1answer
41 views

Even order group, contains element of order 2 [duplicate]

Show every finite group of even order contains an element of order 2. I have tried using Lagrange's theorem, but I am unsure if this is the right path.
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votes
2answers
287 views

Weird isomorphisms of infinite groups

According to my interpretation to one of the answers in Splitting in Short exact sequence, $$\Bbb R \cong \Bbb Q \oplus \Bbb R / \Bbb Q$$ also, according to What is known about the quotient group ...
2
votes
4answers
82 views

If $G$ is a finite group and $H$ is a subgroup of $G$, then $|H| = |gH|$ for every $g \in G$.

I'm stuck. I believe I have half of the proof, but I'm missing an essential part to complete the proof. Any hints would be appreciated. Thanks! Proof: Suppose $G$ is a finite group and $H$ is a ...
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2answers
61 views

Chevron Symbols in Commutator

I am reading a chapter on Commutator in Group Theory and came across chevron symbols "$\langle$" and "$\rangle$" like these: Let $E$ and $F$ be non-empty subsets of $G$, we set $$[E, F] := ...
2
votes
1answer
43 views

Is the question phrased properly? and is my proof correct? (An infinite alternating group is simple)

I'm interested in the following exercise from Dummut & Foote's Abstract algebra text (p. 151) Let $D$ be the subgroup of $S_\Omega$ consisting of permutations which move only a finite number ...
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0answers
23 views

Purely algebraic proof of stable commutator length in a finitely generated solvable group

I've been asked to give a proof that a finitely generated solvable group always has vanishing stable commutator length, as a part of an assignment for an infinite groups course. I think the way to ...
0
votes
1answer
44 views

Sylow $p$-sugroup is normal in a group of order $4p$ [on hold]

Let $p$ is prime and $p\geq 5$. Show that the Sylow $p$-subgroup is normal in a group of order $4p$. Is it true for $p=3$?
3
votes
1answer
25 views

How to prove that predicate is expressible?

I have to prove, that predicate "x is transposition" in $S_5$ group. I can use such symbols, as *, 1, -1, =. However, I don't know any algorithm or way, which can ...
2
votes
2answers
34 views

Proving that the binary operator of a group is commutative if the relation on the group is a partial order

What's probably worst is that this is a repost, as I was stuck on 3. before. The question is this: Throughout this question, we shall denote by $\neg$ the relation on a semigroup $(A, * )$ defined ...
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2answers
52 views

Show that the group is cyclic.

I'm trying to show that the group $U(Z_{54})$ is cyclic. To start, I found the divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54} Then I started to find the elements using the powers of a. Where ...
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0answers
27 views

A question on the symmetric commutator product

Let $G$ be a group and let $R_1,\cdots, R_n$ be subgroups of $G$, where $n\geq 2$. The symmetric commutator product of $R_1,\cdots, R_n$, denoted by $[R_1,\cdots,R_n]_S$, is defined as ...
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1answer
32 views

Which of following is a subgroup of G

Question is todetermine which of following is subgroup of G question stated as I have to check condition $AB^{-1} $ , whether it belongs to H or not , butwhen i check that $Ab^{-1} $ is equal to I ...
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vote
0answers
24 views

Prove that predicate is ineffable

I have a group $(\mathbb{Z}_8,*)$. Also, I have such elements, as *, 1, -1, =. How can I prove, that predicate $g \in \{1,7 \}$ is ineffable?
1
vote
1answer
27 views

Order of $\mathbb{Z}^n/p\mathbb{Z}^n$

I was given the following question: Let $p$ be prime number, and $n$ natural number. I need to calculate the order of $\mathbb{Z}^n/p\mathbb{Z}^n$. I think the answer is $p^n$, but I didn't use the ...
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0answers
22 views

Terminology: how do people call the “normal generating set”?

Let $G$ be a group and let $x_1,\cdots,x_n\in G$ and let $A$ be the normal closure of $\{x_1,\cdots,x_n\}$; that is, the smallest (by inclusion) normal subgroup containing $\{x_1,\cdots,x_n\}$. Notice ...
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vote
1answer
39 views

Prove that $(\{[0],[a],[2a],…,[(b-1)a]\},+)$ is a subgroup of $(\mathbb{Z}_n,+)$.

$n$ is not a prime number. Then $n=ab$ for some $a,b\in \mathbb{N}$: $1<a<n$ and $1<b<n$. Prove that $(\{[0],[a],[2a],...,[(b-1)a]\},+)$ is a subgroup of $(\mathbb{Z}_n,+)$. It seems ...
0
votes
1answer
37 views

Proving a homomorphism $\varphi: F \to G$ with F a free group and $\phi: G \to H$ a surjective homomorphism and $\psi: F \to H$ a homomorphism.

Let $\phi: G \to H$ be a surjective homomorphism and let $\psi: F \to H$ be a homomorphism with $F$ a free group with basis X. Prove that there exists a homomorphism $\varphi: F \to G$ such that $\phi ...
3
votes
2answers
52 views

Is there a more elegant way of proving $\langle (1,2)(3,4), (1,2,3,4,5) \rangle = A_5$

I'm trying to show the following $\langle (1,2)(3,4), (1,2,3,4,5) \rangle = A_5$ I managed to prove this but I think my solution is very inelegant. Here's my argument let $J = \langle ...
0
votes
1answer
69 views

I don't understand this notation- abelian groups

May be a stupid question but is $(\mathbb{Z}^n)_p \equiv \mathbb{Z}^n/(\mathbb{Z}p)^n$ (when $p$ is a prime)??
2
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2answers
41 views

herstein excercise on a finite group

I'm stuck on this herstein exercise for a long time. Let $P$ is a $p$-Sylow subgroup of $G$ and order of $a$ is a prime power then if $a\in N(P)$ prove $a\in P$ I was doing like this but stuck in ...
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0answers
13 views

Mobius transformations — $L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$

$$L^{-1} \times M\lbrace0,1,\infty\rbrace\times L$$ I cannot seem to get the right answer when I multiple them out. $$(1-i)z \cdot (1-z) \cdot \frac{z}{1-i}$$ What do you get when you multiple ...
-1
votes
1answer
30 views

homomorphism/ isomorphism [on hold]

Let $f : G \to H$ be a homomorphism of groups. Let $K$ be a subgroup of $H$, and $A$ a subgroup of $G$. Show that (1) $f^{-1}(K)$ is a subgroup of $G$, (2) $f(A)$ is a subgroup of $H$, (3) if $G$ is ...
6
votes
1answer
106 views
+100

Divisibility of group exponents when the subgroup has finite index.

Let $G$ be a group (not necessary finite) and $H$ a subgroup of $G$ of index $n$ such that exp $(H)<+\infty$ . Show that $$\exp(G)<+\infty$$ and $$\exp(G)\mid\exp(H)\cdot n.$$ Remarks. ...
6
votes
2answers
79 views

Tensor product of (general?) groups

I am starting to learn about tensor products of abelian groups. Why is the tensor product defined for abelian groups? In which part of the construction the commutativity of the groups is needed?
10
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1answer
147 views

Group theoretic solution to an IMO problem

Is there a (strictly) group theoretic interpretation (and possibly a solution) to this problem (taken from the 27th IMO)? "To each vertex of a regular pentagon an integer is assigned in such a way ...
3
votes
2answers
141 views

Group of order 30 can't be simple

I have this following question from my class note on Sylow Theorem: Show that a group of order 30 can not be simple. For that I know the followings: (1) A simple group is one that does not have ...
0
votes
1answer
35 views

Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$

I have this following question from my class note on Sylow Theorem: Let $p$ and $q$ be prime numbers such that $p \nmid (q-1).$ Show that each group of order $pq$ possesses a normal subgroup of ...
4
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0answers
70 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
2
votes
1answer
40 views

Is my proof correct? ($A_n$ is generated by the set of all 3-cycles for $n \geq 3$)

I want to prove that for $n \geq 3$, the alternating group $A_n$ is generated by the set of all 3-cycles. Here is my attempt: Let $\mathcal{S}$ be the set of all 3-cycles in $S_n$, which is a ...
1
vote
1answer
31 views

What does “exponent 2 nilpotency class 2” mean?

According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have ...
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0answers
32 views

finite index subgroups of profinite completions

Let $G$ be a finitely generated, residually finite group, and let $\widehat{G}$ denote its profinite completion. Is there a 1-1 correspondence between finite index subgroups of $G$ and open subgroups ...
1
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1answer
19 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
3
votes
2answers
317 views

If consecutive elements commute each other, does it mean that all of them commutes with each other?

Let $x_1,x_2,...,x_k$ be $k$ different elements of a group $G$ and $k\geq4$. If we know that $x_i$ commutes with $x_{i+1}$ and $x_k$ commutes with $x_1$, can we say that all $x_i$ commutes with each ...
2
votes
2answers
31 views

Problems about generators for Sylow p-subgroups

There are several problems I met asking to find the generators for some different Sylow $p$-subgroup. $(i)$ a Sylow 2-subgroup in $S_{8}$; $(ii)$ a Sylow 3-subgroup in $S_{9}$; $(iii)$ a Sylow ...
4
votes
6answers
210 views

Abelian group of order 99 has a subgroup of order 9

Prove that an abelian group $G$ of order 99 has a subgroup of order 9. I have to prove this, without using Cauchy theorem. I know every basic fact about the order of a group. I've distinguished ...
1
vote
2answers
23 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
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0answers
39 views

$\langle g\rangle$ is $p$-Sylow subgroup of $G_\Delta$?

Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$. If $G$ contains an element of degree and order $p$. $G$ contains the cycle $(1,2 ... p)=g$. Let $\Delta= \lbrace ...
0
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1answer
39 views

Quick question: G-set functor

The Wikipedia page on Representable Functor says: A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds to a ...
0
votes
0answers
29 views

prove that $O^{\pi}(G) \leq K$ . [on hold]

Suppose $G$ is a finite group and $\pi$ will be a set of prime numbers (not empty), if $K \triangleleft \triangleleft G $ ($K$ is subnormal in $G$ ) and $[G:K]$ is a $\pi$-number, then $O^{\pi}(G) ...
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4answers
92 views

Can we conclude that $A= B$? [on hold]

Let $G$ be a group. Suppose that $A\leq B\leq G$ and $[G,A]= [G,B]$. Can we conclude that $A= B$ ?