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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Prove that the map $H → O$ defined by $h → hx$ is a bijection , use this result to deduce Lagrange's Theorem

Exercise: Let $H$ be a subgroup of the finite group $G$ and let $H$ act on $G$ (here $A = G$) by left multiplication. Let $x \in G$ and let $O$ be the orbit of $ x$ under the action $H$. Prove ...
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32 views

Uniqueness of right identity

I'm working on a problem which says the existence of a unique right identity and left inverse (which may not be unique) on a set with binary operation constructs a group. Of course I know that the set ...
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1answer
99 views

Prove the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ is an equivalence relation.

Let $H$ be a group acting of a set $A$. Prove that the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ for some $h \in H$ is an equivalence relation. (For each $x\in A$ the equivalence ...
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2answers
65 views

Problem from “The Theory of Finite Groups” by Kurzweil and Stellmacher

I'm currently working my way through The Theory of Finite Groups by Kurzweil and Stellmacher and came across the following question in the first chapter: Let $G$ be simple, $|G| \ne 2$, and $f : G ...
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0answers
51 views

$ N $ be a minimal normal and $ K $ is a $ p $-nilpotent normal subgroup, then $ [ N , K ] = 1 $

Let $ G $ is a finite group and $ N $ be a minimal normal subgroup of $ G $ whose order is divisible by $ p $, that $ p $ is a prime. Prove that if $ K $ is normal $ p $-nilpotent subgroup of $ G $, ...
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0answers
25 views

Rigid Body Displacements in Group Theory

My professor has defined an RBD (Rigid Body Displacement) of $\mathbb{R}^3$ to be as follows: An RBD of $\mathbb{R}^3$ is an isometry of $\mathbb{R}^3$ such that if $f$ maps a position $A$ onto a ...
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0answers
26 views

Find an infinite group all of whose proper subgroup are finite [duplicate]

Is there an infinite group all of whose proper subgroup are finite? Is there an abelian group satisfying this property?
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0answers
38 views

Smooth admissible representations, Hom, tensor and extension of scalars.

Let $G$ be a locally profinite group, and consider $V$ and $W$ smooth admissible representations of $G$ over some field $F$ (or char. $0$). Let $E/F$ be any field extension. I'd like to find ...
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3answers
49 views

A problem about the alternating group $A_4$

How can I prove that $A_4=\{\sigma^2:\sigma\in S_4\}$? My approach is the following: If $T:S_4\to S_4$ is defined by $T(\sigma)=\sigma^2$, then we have to prove that $T(S_4)=A_4$. Since ...
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1answer
27 views

If $G<S_n$ is transitive, calculate $1/|G| \cdot \sum_{g \in G} f(G)$

$G<S_n$ is transitive calculate $1/|G| * \sum_{g \in G} f(g)$ where $G<S_n$ and $f(g) = |\{ 1 \le i \le n | g(i) = i \}|$ I tried to use the orbit stabiliser theorem but didn't get anywhere ...
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3answers
49 views

If $G$ acts on $\Omega$ and $|G| = |\Omega| + 1$, show there exists nontrivial element fixing point without Burnside's lemma

Let $G$ act on $\Omega$ transitively, and let $|G| = |\Omega| + 1$ (both sets are assumed to be finite). I want to show from first principles (using maybe arguments like the pigeonhole principle, but ...
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3answers
28 views

finding conjugates in group $S_n$

Find the conjugates in $S_3$ of its subgroup <123>. Find conjugate of $A_n$ in $S_n$. my question: i looking for alternative quick way to deal with this questions.thanks.
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1answer
28 views

elements of conjugacy classes of each of $D_4, Q_8 , S_n , A_n $

Enumerate the elements of conjugacy classes of each of $D_4, Q_8 , S_n , A_n $ for n$\le$6. my question: is there a quick way to deal with this question?
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0answers
16 views

Klein 4 group G as a subgroup of $S_4$ under left regular representation. [duplicate]

Q. Represent Klein 4 group G as a subgroup of $S_4$ under left regular representation. my attempt: cauchy theorm states that every group if subgroup of some permutation group. so does K4 but ...
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1answer
21 views

transitive action of G on A.

Assume $G \subset S_A$ is abelian and acts transitively on $A$. Show that $\sigma(a) \ne a$ for any $a \in A$ if $\sigma \ne 1$. Deduce $o(G)=o(A)$. My attempt: Transitively ...
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2answers
210 views

Is a finite group with any subgroup admitting a unique complement, cyclic?

Let $G$ be a finite group such that any subgroup $H \le G$ admits a unique complement $K$. Question: Is $G$ cyclic ?
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28 views

CA-group [abelian centralizer group]

I am searching for all information about CA-groups [abelian centralizer group] and i just found a German book [ Huppert ] and Nilpotent Centralizer group of Suzuki in 44 pages. I need more English ...
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1answer
22 views

Problems with a step in this proof. [generalized quaterion group]

Here is the theorem whose proof I am trying to work through: A finite p-group G having a unique subgroup of order p is either cyclic or generalized quaternion. And here is the part of the proof ...
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1answer
47 views

What are the possibilities that $\sigma\tau\sigma=\tau$ in $S_n$ will be satisfied?

Respected All. Today I post following problem regarding clearing my doubt. I have got the answer. Now after going through the counter example provided by Quang Hoang, now I am willing to establish ...
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2answers
75 views

Group is abelian iff Cayley table is symmetric along its diagonal axis.

The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table is symmetric along its ...
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2answers
70 views

If $\sigma\tau\sigma=\tau$ holds in $S_n$ how to prove/disprove $\sigma, \tau$ are disjoint?

Please help me on the following. I got stuck. We consider the symmetric group $S_n$ of order $n!$. Suppose that $\sigma, \tau$ be two permutation in it satisfying the condition ...
3
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1answer
42 views

Commuting elements with coprime orders, $|xy| = |x||y|$? [duplicate]

Let $G$ be a group. If $x, y \in G$ commute and $\text{gcd}(|x|, |y|) = 1$, does it follow that $|xy| = |x||y|$? EDIT: Progress so far. Let $C = |xy|$; then $x^C = (y^{-1})^C$. I am not sure what do ...
3
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1answer
44 views

$S_n$ contains elements of order $k$ where $k \le n$.

What is the easiest way to see that if $k \le n$, then $S_n$ contains elements of order $k$?
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2answers
40 views

Finite order divides maximal finite order?

Let $G$ be an abelian group, and let $x \in G$ be an element of maximal finite order. If $y \in G$ has finite order in $G$, does it necessarily follow that $|y|$ divides $|x|$? EDIT: Is there a way ...
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5answers
63 views

What is meant by “the coefficients of a polynomial function $f(x)$ are symmetric functions of its roots”?

I am reading Rotman's Introduction to Group Theory. One of his first remarks is that: By the middle of the eighteenth century, it was realized that permutations of the roots of a polynomial ...
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1answer
34 views

Is this answer correct: I'd like some help with Cayley tables

I read this thread here the other day and I also read about Cayley tables that day. As I understood it every column and every row in the Cayley table of the group will contain each element exactly ...
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1answer
35 views

A primitive permutation group with abelian point stabilizars

Suppose that $G$ is a group acting primitively on a set $X$. Also suppose that all of its point stabilizers are abelian. I want to prove that $G$ is either regular of prime degree or a Frobenius ...
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1answer
38 views

Proving two elements of a set are equal based on a two-sided identity

Say I have a set S w/ an associative binary operation *: S x S -> S and a two-sided identity e, and let . Let L and R be elements of S such that L * s = e = s * R How can I prove that L = R ? Since ...
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1answer
35 views

Semidirect product structure of Dihedral group

Let $D_{2n} = \langle x,y : x^n = y^2 =1, yx= x^{-1}y\rangle $ and $H = \langle xy \rangle$. let $n = 2^km$, where $2$ does not divide $m$. Is there any way to write down $D_{2n}$ as the semi direct ...
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1answer
53 views

What does the “free” in *free abelian group* or *free group* mean? [duplicate]

I am asking for the definition of "free". What is the difference between, say, any old abelian group, and a free abelian group?
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problem on dihedral group

i have done 1st and 2nd part. i dont know how to proceed for 3rd part, any help would be appreciated. thanks
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31 views

A question about isomorphism between two groups

Suppose that the group $G$ acts transitively on a set $X$ and $\alpha \in X$. Let $G_\alpha$ be the stabilizer of the action of $G$ on $\alpha$. Also suppose that $$\operatorname{Aut}(X)=\{f:X \to X ...
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1answer
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problem of G action on set A. [closed]

Let a group $G$ acts on a set $A$. Show that if $b=g.a$ for $a,b \in A$ and $g \in G$ then $G_b=g G_a g^{-1}$. find kernel of action if $G$ acts transitively on $A$. Frankly, I have no idea how to ...
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2answers
28 views

prove isomorphisms using first isomorphism theorm

Using first isomorphism theorem, prove the following isomorphisms $\Bbb R/\Bbb Z\xrightarrow\sim S^1,\; $ $\Bbb C/\Bbb R\xrightarrow\sim \Bbb R,\; $ $\Bbb C^\times/\Bbb R^\times_+\xrightarrow\sim ...
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1answer
49 views

How to find out the cardinality of $\mathcal{S}_\beta(S_8)=\{\alpha\in S_8: \alpha\beta\alpha=\beta, |\beta|=2\}$?

I am stuck in the following problem. Please help me. Let us consider the symmetric group $S_n$ of order $n!$. Say $n=8$ viz we are talking about $S_8$. Now let $\alpha, \beta$ be two permutation ...
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35 views

How does any function of a masked value change its distribution?

My question is related to information security, in particular data integrity. Consider a client has a fixed value $y$ and two uniformly random values $a$ and $b$. It computes $v=a\cdot y+b$. Note ...
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33 views

On specific elements in $p$-group

Let $G$ be a non-abelian finite $p$-group, and $x$ is an element of $G$ with maximum possible order, say $p^n$. Then, is it true that $x^{p^{n-1}}$ should be in the center of $G$?
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Sylow subgroups of G of order 240

I'm having some trouble with the following exercise: I am asked to determine how many Sylow $p$-subgroups the group $G$ might have, where $|G|=240=2^4\cdot3\cdot5$. I am not sure how to interpret this ...
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1answer
60 views

Symmetric group, matrix multiplication.

One can associate an $n \times n$ matrix $M_\sigma$ with a permutation $\sigma \in S_n$ by letting the entry at $(i, \sigma(i))$ be $1$ and letting all other entries be $0$. For example, the matrix ...
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2answers
38 views

$l$-Sylow subgroups of $SL_3(\Bbb{F}_p)$ are cyclic for $l|p^2 + p + 1$

Let $l > 3$ be a prime dividing $p^2 + p + 1$, for $p$ a prime number. I wish to prove that all Sylow $l$-subgroups of $G = SL_3(\Bbb{F}_p)$ are cyclic. First, using the surjective group ...
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2answers
41 views

$H \cong K \times H$ implies $K$ is trivial? [duplicate]

Let $H$, $K$ be groups, and suppose that $H \cong K \times H$. Does it necessarily follow that $K$ is trivial?
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Coproduct of $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$ in $\textbf{Grp}$?

Define a group $G$ with two generators $a$, $b$, subject only to the relations $a^2 = e_G$, $b^3 = e_G$. How do I see that $G$ is a coproduct of $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ ...
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1answer
63 views

Binary Operation with Cayley Table

I am asked to write out a Cayley table of a binary operation $\ast$ on the set $$ S = \{1, 2, 3\}$$ for which there is no solution for $\ast$ in $S$ to the equation $1 \ast x = 2$. Here is my ...
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1answer
53 views

How many cosets does the quotient group $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$ have?

How many cosets does the quotient group $\mathbb{Z}_n=\mathbb{Z}/n\mathbb{Z}$ have By definition, a coset is a set composed of all additions obtained by adding each element of a subgroup in turn by ...
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1answer
52 views

Question about order of an element

Suppose $x,y$ are two elements of a finite group $G$ both with order $p$, where $p$ is a prime. And $<x>\neq <y>$. I'm thinking about the order of $xy$. For example, in ...
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Single $\text{GL}_n(\mathbb{C})$-conjugacy class, dimension as algebraic variety?

I have two questions. For each $c \in \mathbb{C}$, is the set$$\Sigma_c = \{A \in M_n(\mathbb{C}) : \text{Tr}(A) = c,\text{ }\text{Rank}(A) = 1\}$$a single $GL_n(\mathbb{C})$-conjugacy class? What ...
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1answer
19 views

$S_k \in Aut(G)$, $deg(G) < (k-1)$?

Given the symmetric group on $k$ symboles, $S_k$, does there exist a connected, undirected simple graph $G$ (no self loops, no multiple edges) of $deg(G) < (k-1)$ such that $S_k \subseteq Aut(G)$? ...
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4answers
437 views

Is there a relationship between vector spaces and fields/rings/groups?

I understand from a comment under Vector Spaces and Groups that every vector space is a group, but not every group is a vector space. Specifically, I would like to know, can I make a statement like: ...
2
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1answer
73 views

Maschke's theorem fails when $p||G|$

Let $G$ be a group and $p$ a prime which divides $|G|$. Let $F$ be a field of characteristic $p$. Let $\epsilon\in F[G]$ - the group algebra - be the sum all elements of $G$. How to show that the ...
4
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1answer
32 views

Group Ring to Hopf Algebra — I'm missing something simple

I've been working through Federico Ardila's online Hopf algebra lectures and hoped to check my understanding so far by constructing the Hopf algebra of a very small group ring from scratch. But I've ...