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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Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$ [duplicate]

I'm working through an Abstract Algebra book to teach myself, and came across the problem: Prove: If $G$ is a finite group of even order, then there exists a $g\in G$ such that $g^2 = e$ and $g ...
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Group Theory Questions-Frattini Subgroup

Given a $p$-group $G$ , and its Frattini subgroup $\Phi(G)$ . How can one prove the following properties: 1) If $H\triangleleft G$ , then $\Phi(G/H)= \Phi(G)/H$ 2) If $G$ is of rank $r$ , and ...
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Is $\{a + bi: a, b \in \mathbb{F}_3\}$ a field?

Is $P = \{a + bi: a, b \in \mathbb{F}_3\}$ a field? This is the question I am having. First, I list the elements in $\mathbb{F}_3$, which consists of $\overline{0}$, $\overline{1}$ and ...
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Finitely generated subgroups of direct limits of groups

Let $G$ be a direct limit of groups $G_n$ for $n\in \mathbb{N}$ (or perhaps even for $n$ in some other directed set, but in my case I only need $\mathbb{N}$). Is it true that every finitely generated ...
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Why cannot the permutation $f^{-1}(1,2,3,5)f$ be even

Please help me to prove that if $f\in S_6$ be arbiotrary permutation so the permutation $f^{-1}(1,2,3,5)f$ cannot be an even permutation. I am sure there is a small thing I am missing it. Thank you.
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How can I conclude that an automorphism is the identity map?

Let $\phi$ be an automorphism on a group $G$. If $\phi$ maps any one non-identity element in $G$ to itself, is $\phi$ necessarily the identity map? What if $G$ is cyclic?
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Let $H \leq G$: prove that $K=\{x\in G:\exists n \in\Bbb{N},\,x^n\in H\}$ is a subgroup of $G$?

$H$ is a subgroup of an abelian group $G$. We define $$K = \{ x \in G : \exists n \in \Bbb{N}, x^n \in H\}.$$ How to prove that $K$ is a subgroup of $G$ ?
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In a group $G$ with operation $\star$, can I apply $\star$ to both sides of an equation?

Theorem: Let $G$ be a group with operation $\star$. For all $a,b,c\in G$, if $a\star b=a\star c$, then $b=c$. I've got a proof, but I'm not sure it is correct (I'm not sure that I can apply the ...
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133 views

What is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})$?

Is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})=\mathbb{Z}/(m\mathbb{Z} +n \mathbb{Z})$? Thanks.
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How to determine which of the elements of a monoid are invertible and the inverses of these elements

Let $*$ denote the binary operation on a set $\mathbb{R}^3$ of ordered triples of real numbers defined such that: $$(A_0,A_1,A_2) * (B_0,B_1,B_2)= (A_0B_0,A_0B_1 + A_1B_0 ,A_0B_2 + A_1B_1 + A_2B_0)$$ ...
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150 views

Finite simple group with subgroups of same order

Let $D$ be a finite simple group with $H < D$ and $K < D$. Also $[D:H]=q$ and $[D:K]=p$, where $p$, $q$ are primes. Want to show that $p=q$. I want to come up with a contradiction with one of ...
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201 views

Prove that a group G such that every element $g \in G$ satisfies the equality $g^2 = 1_G$ is abelian [duplicate]

Possible Duplicate: Prove that if $g^2=e$ for all g in G then G is Abelian. This is how I proved it: Abelian means that the following axioms hold: Associativity, Existence of Identity and ...
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205 views

What are the situations, in which any group of order n is abelian

What are the situations in which any, or particular type group of order n, is abelian ? For example: Group of order $p^2$ is abelian. where p is prime.
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Explanation of first part of proof that if G is any group Z(G) is a normal subgroup of G.

Here's the proof of this fact from Schaum's book on group theory - $1 \in Z(G)$ since $1g = g1$ for all $g \in G$. Consequently $Z(G) \neq \emptyset$. If $g_1,g_2 \in Z(G)$ and $g \in G$, then ...
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Basic Isomorphism Question

The map $\phi: \mathbb{Z} \to \mathbb{Z}$ defined by $\phi (n) = n + 1$ for $n \in \mathbb{Z}$ is one to one and onto $\mathbb{Z}$. Give the definition of a binary operation $*$ on $\mathbb{Z}$ such ...
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Simple method the show that a group of order 15 is cyclic [duplicate]

How can i show with quite simple methods that a group of order 15 is cyclic?
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187 views

Properties of homomorphisms of the additive group of rationals

Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism. Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)? Context The ...
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131 views

$G=\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to?

$G=\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to $S_3$ A subgroup of $S_4$. A proper subgroup of $S_5$ $G$ is not isomorphic to a subgroup of $S_n$ for all $n\ge 3$ What I know is Any finite ...
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Rotation group, altitude

Could someone give me a rigorous proof that the group of rotations each element of which is a composition of rotations around the altitudes of a tetrahedron that transform the tetrahedron into itself ...
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Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\nmid n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
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723 views

p-Sylow groups distinct? [duplicate]

Possible Duplicate: relation between p-sylow subgroups If I have two p-sylow groups $P_{1}, P_{2}$ can I then be sure that $x\in P_{1} \Rightarrow x \notin P_{2}$? That is to say that all ...
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Showing that Ab(G) need not be complete

What's the easiest example to show that $Ab(G)$, the set of Abelian subgroups of a group $G$, need not be complete? I heard that $D_4$ was a good example, but $Ab(D_4)=Sub(D_4)$, which is complete. ...
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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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329 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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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$ 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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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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66 views

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.