A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Group of matrix isomorphism

I have 2 groups: general linear $ k \times k $ with $\cdot$ top-triangle matrix $ n \times n $ with 1 on main diagonal. Operation is $\cdot$ too Is there isomorphism for any any non-trivial ...
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330 views

Are finite indecomposable groups necessarily simple?

The Krull–Schmidt theorem says: If $G$ is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing $G$ as a direct product of finitely many indecomposable ...
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If ord$(a)=m$, ord$(b)=n$ then does there exist $c$ such that ord $(c)=lcm(m,n)$? [duplicate]

Possible Duplicate: Order of elements in abelian groups Let $G$ be an abelian group and suppose that $G$ has elements of orders $m$ and $n$, respectively. Prove that $G$ has an element ...
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525 views

Problem from Armstrong's book, “Groups and Symmetry”

I haven't gotten all that far with this: If $a$, $b$ are members of the permutation group $S_n$, and $ab=ba$, prove that $b$ permutes those integers which are left fixed by $a$. Show that $b$ must ...
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1k views

Using order to show isomorphism in a finite abelian group

How can I show that two finite abelian groups are isomorphic, knowing only that both groups have the same number of elements of any given order? I feel like there should be a nice way to show this ...
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606 views

The commutator subgroup of a quotient in terms of the commutator subgroup and the kernel

Let $H$ be the normal subgroup of $G$. Is it true that $(H[G,G])/H$ is isomorphic to $[G/H,G/H]$? If so, I want to make a surjective homomorphism $\phi\colon H[G,G]\to [G/H,G/H]$ with Kernel $H$ to ...
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547 views

Quotient of an Abelian group with its torsion subgroup

Let $A$ be a finitely generated Abelian group. Let $tA$ denote the torsion subgroup. Prove that $A$ has a subgroup isomorphic to $A/tA$. I know that $A/tA$ is torsion free, so my thinking so far has ...
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46 views

How do I proceed with this proof about order of elements in a group G?

Let a and b be elements of a group G. Let ord(a)=m and ord(b)=n. Prove if m and n are relatively prime, then no power of a can be equal to any power of b (except for e). My work so far: Suppose m ...
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57 views

Explain why D3 cannot be a subgroup of D8

To be a subgroup, a subset of a group must satisfy the group axiom but in this case, I do not see how the group axiom plays a part. Could someone explain to me why the above question is true?
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38 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
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Need help determining what this center is isomorphic to

I am looking at this released exam, problem 2c. It states: Let $G$ be a non-abelian group of order $8$ and $Z$ be the center of $G$. To which group is $Z$ isomorphic? It gives a hint to recall a ...
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49 views

Basic doubt about cosets

Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space ...
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53 views

Let $G$ be a group and $u \in G$ be a fixed element. By the following, prove that $(G,\bullet)$ is a group.

Let $G$ be a group and $u \in G$ be a fixed element. Define the operation $\bullet$ on G as $\forall a,b \in G, a \bullet b=au^{-1}b.$ Prove that $(G,\bullet)$ is a group. So, I know that in order ...
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155 views

Two problems in Group theorem related to Sylow's theorem(maybe)

Prove that any subgroup of order $ p^{n-1} $ in a group $G$ of order $p^{n}$, p a prime number, is normal in $G$. $(a)$ Prove that a group of order 28 has a normal subgroup of order 7. To deal ...
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62 views

Mapping Normalizer to Automorphism

I am squeezing my brain trying to understand this problem: Let H be a subgroup of $G$, and denoted by $\operatorname{Aut}(H)$ the group of all automorphisms of $H. $ Define $$\alpha : N_G(H) ...
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245 views

How many ways can the vertices of an equilateral triangle be colored using three different colors?

Its not $3^3$ because some of the colorings are equivalent. How would I apply Burnside's theorem to this?
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60 views

Let $G$ be a group. If $H\leq G$ is a subgroup and $N\vartriangleleft G$, then $HN$ is a subgroup of $G$.

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. Prove that $HN=\{hn\:|\:h \in H, n \in N\}$ is a subgroup. Here what I have so far. It is not really much I understand what I need to ...
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92 views

Proving that the order of a group is greater than or equal to the product of orders of 2 subgroups.

Let $H$ and $K$ be subgroups of a finite group $G$ such that $H \cap K = \{e\}$ Prove that $|G|\ge |H||K|$ What I think is the correct step is to consider the cosets $hK, h \in H$, and then using ...
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51 views

Existence of homomorphism between two groups

Can there exist an onto group homomorphism from $S_5$ to $ S_4$ or from $S_5$ to $\mathbb Z_5$?Is it possible to write the homomorphism explicitly?
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Order of the group.

An abelian group $G$ is generated by $x$ and $y$ with $$O(x)=16,O(y)=24,x^2=y^3$$ What is the order of $G$? My attempt:There are $24+16-1=39$ elements generated by $x$ and $y$ separately. Also ...
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102 views

Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?

Is the following statement true or false: If $G$ is a group with the property that $g=g^{-1}$ for all $g \in G$, then $G$ is abelian. I believe it is false since I know that abelian ...
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171 views

Other method(s) to prove “a group cannot have exactly two elements of order $2$”

If $a,b$ are elements of a group having order $2$ then, if $a,b$ commute, $ab(\ne a , \ne b)$ has order $2$, and if $a,b$ do not commute, then $aba^{-1}(\ne a , \ne b, \ne e)$ has order $2$. Using ...
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72 views

Does there exists a homomorphism for any groups $G$ and $H$

This is a question from Exercise 8.2 of Visual Group Theory which says:determine whether true or false. For any group $H$ and $G$,there is some homomorphism from $H$ to $G$. For any groups $H$ and ...
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111 views

Image of centralizer under an isomorphism

Suppose we have a group isomorphism $\phi: G\rightarrow K$ between two finite groups and let $H$ a subgroup of $G$. Are there any known facts about the image of the centralizer $C_G(H)$ of $H$ in $G$ ...
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72 views

$ o(ab)= o(ba)$, What about when $ab$ has infinite order?

I have been able to prove this when I say simply assume $o(ab)= n$, but I have been pondering the case of where $o(ab)$ is infinite? Is this something that needs to be considered? And if not, why? ...
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Show if $\pi,\sigma$ are any permutation s.t $(\pi\sigma)^2=\pi^2\sigma^2$, then $\pi\sigma=\sigma\pi$…

As the title says, the problem is: Show if $\pi,\sigma$ are any permutation s.t $(\pi\sigma)^2=\pi^2\sigma^2$, then $\pi\sigma=\sigma\pi$. There is a theorem that states If $\pi,\sigma$ are ...
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69 views

Maximal subgroups of $S_4$

I have to prove that $$ \operatorname{Frat}(S_4):=\bigcap_{M\stackrel{\max}{\le} S_4}M=1 $$ but I don't know how to compute it since I don't know what are the maximal subgroups in $S_4$. EDIT: ...
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Calculating $x^{103} \equiv 2 \pmod{143}$

I need to find x, given that: $0\leq x \leq 143$ and $x^{103}\equiv 2 \pmod{143}$. I tried to use Euler's theorem $p(143)=120$, but it didn't help.
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triangle groups

I am having a hard time finding references(apart from wikipedia) for the geometric interpretation of triangle groups $$T_{a,b,c} =\langle x,y: \, |x|=a,|y|=b,|xy|=c \rangle.$$ How can these groups be ...
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If $xy=x^{-1}y^{-1}$, does this imply $x=x^{-1}$

This seems like a simple enough question, trying to show that if the title condition holds, that a group $G$ of which $x,y$ are elements, then $G$ is Abelian. $$xy=x^{-1}y^{-1}=(yx)^{-1}$$ From here I ...
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135 views

Quotient groups and normal subgroups

I am wondering if there is some characterization of the normal subgroups of a quotient group. More precisely let $G$ be a group and $H$ a normal subgroup. Let $U$ be a normal subgroup of the quotient ...
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I don't understand equivalence relations

I'm trying to show that the left coset with a subgroup is an equivalence relation. So taking some element $g \in G$ and a subgroup $H$, the left coset is defined as $gH = \{gh : h \in H\}$. That means ...
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100 views

Let $G$ be the Klein 4-group. What four permutations in $S_4$ form a subgroup $\cong G$?

I know that $G$ is an abelian group and is composed of $\{e, a, b, ab\}$. I also know that every non-identity element has order 2. But I'm not sure how to write out the permutations $F_e, F_a, F_{ab}$ ...
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Is the group of all determinants of all invertible $n \times n$-matrices isomorphic to $\langle\mathbb{R}^*, \cdot\rangle$?

I am doing Linear Algebra and Abstract Algebra simultaneously, and in Linear Algebra class, going through determinants, I thought of something interesting (for a freshman just learning the subjects, ...
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Abelian groups of finite order

Show that an abelian group of order 75 has a cyclic subgroup of order 15. Do I need to use the fundamental theorem of finite abelian groups in some way?
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108 views

Basic Abstract Algebra - Subgroups of Abelian Group

I'm trying to prove the following: Let $G$ be an abelian group of order 72. Show that $G$ has exactly one subgroup of order 8. I know by theorem that $G$ must have at least one subgroup of order ...
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95 views

No group of following property. Is this true?

Let $p$ be a prime greater than 3 and $G$ be group of order $p^5$. Is it true that there is no group $G$ of order $p^5$ such that the order of frattini subgroup is $p^3$ and the order of center is ...
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229 views

Let $G$ be a finite group and $p$ be a prime. Suppose that every proper subgroup of $G$ of order divisible by $p$ is a $p$-group.

Let $G$ be a finite group and $p$ be a prime. Suppose that every proper subgroup of $G$ of order divisible by $p$ is a $p$-group. Prove that every two distinct Sylow $p$-subgroup of $G$ intersect ...
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60 views

If an element $c$ of order $d$ belongs to both $\langle a\rangle$ and $\langle b\rangle$, then $\langle a\rangle=\langle b\rangle=\langle c\rangle$?

If an element $c$ in a group $G$ belongs to both $\langle a\rangle$ and $\langle b\rangle $ (where $a$, $b$ belong to $G$), then if $|a|=|b|=|c|=d$, prove that $\langle a\rangle =\langle b\rangle ...
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Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
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Non-trivial nilpotent group has non-trivial center

A book I'm reading quotes the following result without any explanation: Any non-trivial nilpotent group has a non-trivial center. (The definition of "nilpotent group" is as follows: Suppose $G$ is a ...
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191 views

Index of conjugate subgroup equals the index of subgroup

Show that if $G$ is a group and $H$ is its subgroup then $[G:H]=[G:gHg^{-1}]$, $g \in G$. Attempted solution: Let $f:G\mapsto\hat{G}$ be a group homomorphism such that $\mbox{Ker}f \subseteq H$ we ...
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66 views

Prove that $N$ is normal

Let $H$ be a subgroup of $G$. Consider the set $N=\cap_{x\in G}xHx^{-1}$. Prove that $N$ is normal subgroup of $G$. Using the fact that any (finite or infinite) intersection of subgroups is a ...
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52 views

Condition for $\langle m\rangle\subseteq\langle n\rangle$

Assuming $m,n\in{\mathbb{Z}}$, what is the necessary and sufficient condition such that the question at hand is valid? If $\langle m\rangle=\{m^a|a\in\mathbb{Z}\}$ and $\langle ...
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320 views

Infinite coproduct of abelian groups

One can see on every text (book, lesson, comments) that a direct sum/coproduct of abelian groups is the same as a finite product but in the infinite case, the direct sum/coproduct is only a subgroup ...
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203 views

Can someone explain automorphisms to me?

So I know the definition of an automorphism is an isomorphism that maps from a group to itself. How can an element of an automorphism map to something besides itself?
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137 views

Why are all cyclic groups countable?

I'm new to group theory. Why are all cyclic groups countable? And does countable mean finite or denumerable?
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101 views

Intersection of subgroups

Let $A,B$ be subgroups of a group $G$. We define: $$C=\{x; x \in A \wedge x \in B \}$$ Prove that $C$ is a subgroup of $G$ My attempt at a proof: $1)$ Closure: Since A and B are subgroups: $$x \in ...
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284 views

the number of the subgroups of a non cyclic group whose order is $25$

My question is about group theory: How many subgroups does a non-cyclic group contain whose order is 25? How can i answer that question? Can you generalize the answer? Thanks for your help.
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158 views

simple group theory question

Can someone give an example of a finite (ideally nonabelian) group $G$ and two surjective homomorphisms $\phi_1,\phi_2 : F_2 \rightarrow G$ (where $F_2$ is the free group on the generators $x,y$), ...