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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There cannot be a group of units of order $14$

I proved the following claim: There is no $n$ such that $U(n)$ has order $14$. Please could someone tell me if my proof is correct? $U(n)$ is the group of units modulo $n$ and $\varphi$ is the ...
2
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1answer
76 views

Short exact sequence of groups, is it possible to construct an associated fibration of spaces?

Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, 1) \to K(F, 1) \to K(G, 1)?$$
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1answer
44 views

Convincing normal subgroup proof?

I wrote the following proof on an exam, I was wondering if it makes sense. Question: let $H$ be a subgroup of $G$, written in multiplicitive notation. Prove that if the coset multiplication defined ...
2
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1answer
40 views

Let $G$ be a $2$-group and suppose the centralizer of some element of order two has order at most four, then $G$ has maximal class

Let $G$ be a $2$-group of order $|G| \ge 4$ and $H \le G$ be a subgroup of order $2$, i.e. $|H| = 2$. Suppose we have $|C_G(H)| \le 4$. Then $G$ has maximal class. Do you know a proof of this fact? I ...
2
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1answer
61 views

Normalizers of subgroups of Sylow $p$-subgroups

I am wondering whether there is an easy example of a finite group $G$ with a Sylow $p$-subgroup $P$ and a subgroup $Q\leq P$ such that the normalizer $N_P(Q)$ of $Q$ in $P$ is NOT a Sylow $p$-subgroup ...
2
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1answer
57 views

For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
2
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1answer
31 views

Why $U(p^m) \oplus U(q^n)$ is not cyclic

I tried to solve the following exercise: Let $p,q$ be odd primes and $n,m$ positive integers. Explain why $U(p^m) \oplus U(q^n)$ is not cyclic. I solved the question as follows: We have $U(p^m) \...
2
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1answer
23 views

Regarding part of proof of proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.

Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space. Part of Proof: Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity element). ...
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1answer
46 views

Is the definition given by the GAP-manual equivalent to the one given in the site? [closed]

Here http://groupprops.subwiki.org/wiki/Normalizer_of_a_subset_of_a_group the normalizer of a subset of a group is defined. GAP gives the following description of the Function IsNormal : 39.3.6 ...
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29 views

Help showing $H_n=\sum_{h \in H}h \in F[H]$ is a simple submodule

Given $F$ a field and $H$ a finite group and $H_n=\sum_{h \in H}h \in F[H]$, I'm trying to show that the left ideal $(H_n)\subset F[H]$ is a simple submodule. Now I'm not really sure what the ideal ...
2
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1answer
35 views

A question on finitely generated Abelian groups with a minimal number of generators

In my class on group theory I have encountered this strange looking question relating to Abelian groups in terms of generators which states: We are to find, up to isomorphism, all Abelian groups $...
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4answers
98 views

How to use a proof by contradiction in group theory!

A group $G$ is said to be abelian-by-finite if it has a normal subgroup $H$ of finite index in $G$. I want to prove the following statement: "If a group $G$ has property $\mathcal P$ then it is ...
2
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1answer
32 views

Restriction of automorphism to the generalized Fitting subgroup

for my master's thesis I am going through M.Hertweck and W.Kimmerle's article on Coleman automorphisms. I encountered the following reasoning, which I can't seem to follow. We know the following on ...
2
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2answers
44 views

When is every group of order $n$ nilpotent of class $\leq c$?

In the paper 'Nilpotent Numbers' by Pakianathan and Shankar (http://www2.math.ou.edu/~shankar/papers/nil2.pdf), it was proven that every group of order $n$ is nilpotent if and only if $p^k\not\equiv 1\...
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1answer
29 views

Group works on topological space

I have to prove that if discrete and finite group G works on topological Hausdorff space X $ \varphi :G \times X \rightarrow X $ and $ \varphi $ is cotinuous function, then $ X / G $ is also a ...
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1answer
30 views

Show that every $\sigma \in S_n$ is of the form $\sigma = \prod_i (1 \; \; x_i)$

Let $n \in \Bbb N$ and let $S_n$ denote the group of permutations of $\{1,2,...,n\}$. Prove that for all $\sigma \in S_n$, we have: $$\sigma = \prod_{i=1}^m (1 \ \ x_i), \text{ for some $x_1,....
2
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1answer
58 views

The equation $|A^{-1}A|=|A|^2-|A|+1$ for finite subsets of a group

Let $A$ be a finite subset of a group $G$. It is clear that $|A^{-1}A|\leq|A|^2-|A|+1$. If $A$ is singleton or $A=\{1,a\}$ with $O(a)\neq 2$ then the equality holds. Now, can somebody characterize all ...
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1answer
19 views

Z modules spanned by row space of matrix invariant under matrix multiplication

I have met this strange looking problem on which I have no idea, from my course on Abstract Algebra dealing with modules: Let $ v_1,...,v_k \in \mathbb{Z}^n $ row vectors of length n over $ \...
2
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1answer
64 views

Give an Example of a Bijective Function $\mathbb{Z} \rightarrow \mathbb{Z}$ with precisely $r$ orbits

This problem is a continuation of what was discussed here. As the title states, I wish to find a bijective function $\mathbb{Z} \rightarrow \mathbb{Z}$ with precisely $r$ orbits. I observe from the ...
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2answers
33 views

Changing rules of multiplication

I'm working on a problem where I would like to change the rules of multiplication. Suppose I am multiplying $N$ numbers: $$X = A_1 A_2 A_3 \dots A_N$$ I would like the magnitude of $X$ to be the ...
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2answers
40 views

Proving that a subset of a ring $R$ is a subring

In this example, $R$ is a ring with unity $1$, with $a\in R$ having the property $a^2=a$ (making it a Boolean ring). I know every Boolean ring is of characteristic 2 since: $a+a=(a+a)^2=a^2+a^2+a^2+a^...
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1answer
32 views

If $A \lhd P$ and $A = C_P(A)$, then $|P:A|$ divides $(|A| - 1)!$

This is problem 1.D.10 in Isaacs, Finite Group Theory. Let $A$ be maximal among the abelian normal subgroups of a $p$-group $P$. Show that $A = C_P(A)$, and deduce that $|P:A|$ divides $(|A|-1)!$ ...
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1answer
35 views

Conjugacy classes of a group [closed]

What is number of conjugacy classes in the permutation group $S_{6}$ I only knows that $o(S_6)=6!$
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1answer
54 views

The value of the rational “Möbius”-like transformations at infinity

If $\mathbb F$ is some field, the group $PSL(2, \mathbb F)$ consists of the mappings $$ x \mapsto \frac{ax + b}{cx + d} $$ with $a,b,c,d \in \mathbb F$ and $ad - bc = 1$. These mappings are defined ...
2
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1answer
51 views

When the automorphism group is trivial… [duplicate]

If $G$ is a trivial group , obviously $Aut(G)$ is trivial . Does the converse hold $?$ . When we are given that a group $G$ has trivial automorphism group , can we conclude that the ...
2
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1answer
43 views

Why are these two groups isomorphic?

$\langle x, y \mid x^7 = e, y^3 = e, yxy^{-1} = x^2 \rangle$ and $\langle x, y \mid x^7 = e, y^3 = e, yxy^{-1} = x^4 \rangle$. Is it because $x^4 = (x^2)^2$, or is this the wrong reason why they are ...
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1answer
90 views

How to show $\hat{\mathbb{Z}}/n\hat{\mathbb{Z}}\cong \mathbb{Z}/n\mathbb{Z}$

I'm trying to do exercise 1.9 from the following PDF: http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf RELATED: Elements in $\hat{\mathbb{Z}}$, the profinite completion of the ...
2
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2answers
215 views

Group of order $255$ is cyclic

Let $G$ a group and its order is $255$. Prove that $G$ is cyclic. I easily demonstrated that the group has only one $17$-Sylow subgroup $P$ that is normal in $G$ and it's cyclic since it is of a ...
2
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1answer
46 views

Conjugacy of subgroups of free groups

Let $F(X)$ be the free group on a finite set $X$. Suppose that $H \subseteq F(X)$ is a subgroup where $[F(X):H]$ is finite. What can we say about the size of $[F(X):N(H)]$ where $N(H)$ is the ...
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2answers
34 views

Centralizer of a equal to generator of for a nonabelian group $G$ order $pq$

Let $p$ and $q$ be odd primes such that $p < q$. $G$ is nonabelian group of order $pq$. Prove if $a \in G$ and isn't the identity, then $\langle a \rangle = C(a)$. So I was able to prove $\...
2
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1answer
54 views

Two definitions of nilpotence

I don't understand why the following two definitions of nilpotence are equivalent: Definition 1. $G$ is $0$-step nilpotent if $G=\{e\}$. G is $k+1$th-step nilpotent if G is not $k$-step nilpotent, ...
2
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2answers
44 views

How to prove $\operatorname{ord}_G(a^k) = r/\gcd(r,k)$?

If $G$ is a finite group and $a \in G$ an element with $\operatorname{ord}_G(a) = r$, then $\operatorname{ord}_G(a^k) = r/\gcd(r,k)$. I know that this statement is correct, but how can one prove it?
2
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1answer
64 views

$H \leq G$ s.t. $|G : H|$ is the least prime dividing $|G|$. Show $H$ is normal in $G$. [duplicate]

Let $G$ be a finite group, $H \leq G$ such that $[G : H]$ is the least prime which divides $|G|$. Show that $H$ is normal in $G$. Can someone explain what information we get from knowing that $[G : H]...
2
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1answer
40 views

Group action on Finite Field

Suppose $F=\mathbb{F}_{p^n}$ is a degree-$n$ extension of $\mathbb{F}_p$. My questions concerns the action of the multiplicative group $(\mathbb{F}_p)^{\times}$ on $F$ by left multiplication. we can ...
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2answers
47 views

Is there a non abelian group that characterize a one dimensional lattice structure?

Of all groups that characterize a one dimensional lattice structure (symmetry operations including translation, $C_2$, mirror plane, inversion point), is there a non abelian one? Moreover, Can it have ...
2
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2answers
96 views

If $|a^{2}|=|b^{2}|$ then $|a|=|b|?$

If $|a^{2}|=|b^{2}|$ (for non identity elements $a$ and $b$ of a group $G$ and $|a|$ denotes the order of the element $a$) prove or disprove that $|a|=|b|.$ I tried as follows Clearly infinite order ...
2
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1answer
37 views

What should be isomorphism class of $Aut(\mathbb{Z}_{p^a}\times \mathbb{Z}_{p^b q^c})$?

I am trying to get some ideas on the following problem but no result. Please show me the way. Aut$(G)$ denotes the group of automorphisms of the group $G$. If $p, q$ are distinct primes then how ...
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3answers
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Quotient ring $\mathbb{Z}/4\mathbb{Z}$

I'm trying to understand what the quotient ring is. I know that $\mathbb{Z}/4\mathbb{Z} = \mathbb{Z}_4$, but I can't get the same result by myself. Having used the definition of the quotient ring that ...
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1answer
64 views

Universal Property of the Grothendieck Group (Group Completion)

I am following the (german) construction of the Grothendieck group at https://de.wikipedia.org/wiki/Grothendieck-Gruppe. It is very precise and easy to follow until to the point of the universal ...
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2answers
43 views

Automorphism and conjugation

Let $G$ be a group and fix $h\in G$. Show that $f:G\to G$ given by $f(g)=h^{-1}gh$ is an automorphism. Do I need to show that $f$ is bijective first? And then relate it to being an automorphism? A ...
2
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1answer
47 views

If $C_G(x) \leq H$ for every $p$-element $x \in H$, then $p$ cannot divide both $|H|$ and $|G:H|$

This is problem 1.D.2 in Isaacs, Finite Group Theory. I am self-studying, so would appreciate a proof verification. Note: in this book, all groups are assumed finite unless otherwise stated. Fix ...
2
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1answer
24 views

Some properties of the largest abelian $p$-factor group and its kernel $A^p(G)$.

Let $G$ be a finite group and let $A^p(G)$ be the unique smallest normal subgroup of $G$ for which the corresponding factor group is an abelian $p$-group (that $A^p(G)$ is well-defined is an immediate ...
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1answer
53 views

Confused about this exercise question: if a set with a certain binary operation is a group

I tried to answer the following exercise: Let $S$ be a nonempty set with an associative operation that is left and right cancellative. Assume that for every $a$ in $S$ the set $\{a^n \mid n=1,2,3, \...
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1answer
40 views

Is it possible for a group to be a finite union of subgroups of infinite index?

Just restating the title: Does there exist a group $G$ and subgroups $H_1, \ldots, H_k$ so that $[G:H_i]$ is infinite for each $i = 1, 2, \ldots, k$, and $G = H_1 \cup \cdots \cup H_k$? If $G$ is a ...
2
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1answer
33 views

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime.

Let $G$ be a group and show that $|G:Z(G)|$ cannot be prime where $Z(G)$ is the center of $G$ and $|G:Z(G)|$ is the index of $Z(G)$ in $G$. This was in a test that I had recently but I was not able ...
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1answer
37 views

Prove the following set endowed with the binary operation is an abelian group

Let $∗$ be a commutative and associative binary operation on a set $S$. Assume that for every $x$ and $y$ in $S$, there exists $z$ in $S$ such that $x ∗ z = y$. Show that this set with the operation $∗...
2
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1answer
69 views

intersection of all maximal centralizer

Let $G$ be non-abelian. Let say $C_G(x)$ is maximal centralizer if $x\in G-Z(G)$ and there is no $g\in G-Z(G)$ such that $C_G(x)< C_G(g)$. Now let $C$ be the intersection of all maximal ...
2
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1answer
51 views

Does $H_1/\ker(\phi_{|H_1})=H_2/\ker(\phi_{|H_2})$ imply that $H_1=H_2$?

Let $\phi: G \rightarrow G'$ be a group homomorphism that is onto and $H_1 \subseteq H_2$ are subgroups of $G$. In addition, $\phi(H_1)=\phi(H_2)$ and $\ker(\phi_{|H_1})=\ker(\phi_{|H_2})$. I suspect ...
2
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1answer
51 views

Rotations of 4-Cubes

I have recently learned the orbit stabilizer theorem, and have encountered unexpected results pertaining to the rotations of a tesseract; I am curious if there is any intuition for this. A $4$-Cube ...
2
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1answer
34 views

Why don't Archimedean solids give finite subgroups of $SO(3,\Bbb R)$?

I know that the Platonic solids correspond to finite subgroups of $SO(3,\Bbb R)$. For example, the tetrahedron corresponds to a subgroup isomorphic to $A_4$. The cube and octahedron to one isomorphic ...