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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A proof regarding finite abelian groups and their orders.

Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Given $G=H\times K$, where $H=\{x\in G|x^{p^n}=e\}$ and $K=\{x\in G|x^m=e\}$. We need to prove that ...
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31 views

Are there elements of order of order $4$ in the group $(\mathbb{Z}_{30}, +)$?

Are there elements of order of order $4$ in the group $(\mathbb{Z}_{30}, +)$? What is a way to approach this question without Lagrange's Theorem?
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36 views

Isomorphism between vector space and translation group

Hi Im trying to find an isomorphism between the group of translations of a vector space $V$ and the vector space itself with the addition $(V,+)$ In the exercise it is not precised if the addition $+$ ...
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0answers
46 views

Lie Bracket on Lie Algebra vs. Lie derivative on corresponding group orbit tangent vector field

Suppose I have a matrix group, for example the 2-d affine group and two elements of Lie algebra, A and B, expressed as 3x3 matrices. The Lie bracket is [A,B]=AB-BA. Suppose on the other hand I ...
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41 views

$G$ has exactly $n$ different subgroups of prime order $p$, then total number of elements of $p$ is $m(p-1)$

$A$ and $B$ are two distinct subgroups of group $G$ s.t. $o(A)=o(B)=p=prime$, show that $A\cap B=\{e\}$. Deduce that if $G$ has exactly $n$ different subgroups of prime order $p$, then total ...
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16 views

Construct a digraph (direct graph ) on n vertices

I want to contruct a digraph based intruction below : For each n≥2, construct a digraph on n vertices such that 〖deg〗^+ (v)≠〖deg〗^+ (u) and 〖deg〗^- (v)≠〖deg〗^- (u) for all vertices v≠u. I know how ...
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90 views

How do we compute Aut(Z2 x Z2)?

I know that $Aut(Z_n)=u(n)$. While computing $Aut(Z_4\times Z_2)=Z_2$ and $Aut(Z_2\times Z_2\times Z_2)=S_3$, I considered all possible cases and checked if it was a homomorphism. I am not sure if ...
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2answers
42 views

Let $G$ be a group with $33$ elements. If $H$ and $K$ are subgroups of $G$ of order $3$ and $11$ then show that $G = HK$.

Let $G$ be a group with $33$ elements. If $H$ and $K$ are subgroups of $G$ of order $3$ and $11$ then show that $G = HK$. I only know that if $H$ and $K$ are subgroups of $G$ then $HK$ is defined ...
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51 views

Let $G$ be a cyclic group of order $n$. Prove that every subgroup $H$ of $G$ is of the form $<a^m>$ where $m$ is a divisor of $n$.

Let $G$ be a cyclic group of order $n$. Prove that every subgroup $H$ of $G$ is of the form $<a^m>$ where $m$ is a divisor of $n$. We have $o(a^m)=\frac{n}{gcd(m,n)}=\frac{n}{m}, ...
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1answer
33 views

automorphism of abelian grp. of prime power order

I had an exercise to be proved: Assume we have an abelian group $G$ whose order is a power of prime $p$, and $H$ is its only subgroup of order $p$ . let $\phi:G\rightarrow G$ be the ...
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2answers
35 views

Prove that the size of every conjugacy class of a finite group divides the order of the group

We are effectively asked to show that a map is a bijection. $$\phi: (G:C_G(a))\rightarrow (a)$$ $$ C_G(a)x \rightarrow x^{-1}ax$$ Need to show that the above is well-defined, 1-1 and onto first and ...
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1answer
28 views

Let $H$ be a group. Let $a, b$ be fixed positive integers and $H=\{ax+by\mid x,y\in \Bbb Z\}.$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$.

Let $H$ be a group. Let $a, b$ be fixed positive integers and $$H=\{ax+by\mid x,y\in \Bbb Z\}.$$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$. Given $d=\gcd(a,b)$ then $d|a, ~d|b$ i.e. $d\alpha ...
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130 views

How to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$ [closed]

I'm having trouble trying to find composition series for $GL_{2}(\mathbb{Z}/5\mathbb{Z})$ and $GL_{2}(\mathbb{Z}/25\mathbb{Z})$. I have no idea how to construct this... Someone says that the number ...
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1answer
75 views

Let $ T/N = \Phi(G/N) $. Why $ K \cap T = 1 $?

Let $ G $ be a soluble group whit $ \Phi(G) = 1 $ that $ \Phi(G) $ is a Frattini subgroup of $ G $. Let $ K $ and $ N $ are minimal normal subgroup of $ G $ that $ K \neq N $. Let $ T/N = \Phi(G/N) $. ...
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1answer
41 views

$ G$ be a group of order $30$ generated by $a$.

Let G be a group of order $30$ generated by $a$. Then (i) find the order of the cyclic subgroup generated by $a^{18}$ (ii) Find the subgroup $H$ of order $6$. Find the generator of $H$. ...
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2answers
91 views

Generator of $S_n$

I know that for $n\geq2$ $S_n$ (the symmetric group of $n$ symbols) can be generated by only two elements, among which one is a $n$-cycle and other one is a transposition. But is it true that for any ...
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32 views

Permutation group notation.

This is from the wikipedia page about parity of permutations. I'm having a little trouble understanding why $(1 3 5)(2 4)$ is equal to $(1 5)(1 3)(2 4)$. I know they are both supposed to take ...
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13 views

Left invariance of a differential operator

Given $x, a\in G$ with G a group and x and a fixed, does the left invariance of a differential operator D on G imply that $D[f(a^{-1}x)]=D[a^{-1}f(x)]$?
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1answer
89 views

Prove that a non-abelian group of order $8$ must have an element of order $4$.

Prove that a non-abelian group of order $8$ must have an element of order $4$. To solve it, one can use the concept upto Lagrange's th. Attempt: We have $o(element)|o(Group)$ then here order ...
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1answer
27 views

Let a, b be fixed positive integers and $H=\{ax+by/x,y\in \Bbb Z\}.$ Show that $H$ is a cyclic group with $gcd(a,b)$ as a generator.

Let a, b be fixed positive integers and $$H=\{ax+by/x,y\in \Bbb Z\}.$$ Show that $H$ is a cyclic group with $gcd(a,b)$ as a generator. Approach: Let $d=gcd(a,b)$ then $d|a, ~d|b$ i.e $d\alpha ...
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1answer
56 views

Infinite torsion CAT(0) groups

Do all infinite CAT(0) groups contain a $\mathbb{Z}$ subgroup? I am aware that this has been established for hyperbolic groups, and similar questions have appeared on open questions lists for CAT(0) ...
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1answer
37 views

$2$-Sylow subgroup of nonabelian group of order $56$ must be abelian?

Let $G$ be a nonabelian group of order $56$. Let $ Q$ be the $2$-Sylow subgroup of $G$. Show that $Q$ is isomorphic to $\mathbb Z_8, \mathbb Z_4\times \mathbb Z_2$ or $\mathbb Z_2\times \mathbb ...
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52 views

A question about the proof of a theorem in representation theory

My question is about some parts of the proof of Theorem 8.1.10 from the book "A Course in the Theory of Groups" by Derek J.S. Robinson. To prove Theorem 8.1.10, we want to prove that there is a ...
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0answers
38 views

Size of group of roots of unity

Let $p$ be an odd prime and $G=(\mathbb{Z}/p^N\mathbb{Z})^{\times}$. Let $a=p^{N-1}$, $b=p-1$, $A=\{g\in G: g^{a}=1\}$, $B=\{g\in G: g^{b}=1\}$. Prove that $A,B$ are subgroups of size $a,b$ ...
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1answer
17 views

Argument regarding generators of S3

In my notes, a certain argument goes: Let $H=<(1,2)>$, $K=<(2,3)>$, by the formula $|HK|=2.2/1=4$, we know that $HK$ is not a subgroup of $S_3$ by Lagrange's Theorem. So far I can follow ...
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8 views

Symmetry of the multiplier of a l.c.s.c. abelian group

Let $G$ be a l.c.s.c. (locally compact, second countable) Abelian group, and let $\hat{G}$ be its (well-defined) dual. Consider the group $G\oplus\hat{G}$ (which is a l.c.s.c. Abelian group itself), ...
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50 views

No finitely generated abelian group is divisible

My question might be ridiculously easy. But I want prove No finitely generated abelian group is divisible. Let $G$ be a finitely generated abelian group. By definition, group $G$ is divisible ...
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50 views

Without using Cauchy's theorem: If $G$ an abelian group of order $10$ contains an element of order $5$, show that $G$ must be a cyclic group.

If $G$ an abelian group of order $10$ contains an element of order $5$, without using Cauchy's theorem, show that $G$ must be a cyclic group. In my book, there is a hints that $G$ has an element ...
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1answer
18 views

Isomorphisms of subgroups of dihedral groups.

I think I have the correct answer but I just want to double check. I think one of the solutions is {e,$r^2$,$r^4$,j,$r^2$*j*,$r^4$*j*}. I believe the second one is {e, $r^2$,$r^4$,r j,$r^3$j,$r^5$j} ...
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59 views

Is the symmetric group $S_4$ cyclic

Is the symmetric group $S_4$ cyclic? By writing all $24$ elements we can write the tabular form of $S_4$. Then choosing each element of $S_4$, we can find its order and thus, we can show that ...
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26 views

Commuting elements in ${\rm PSL}(2,\mathbb{Z})$ are powers of some element

In free groups, it is easy to show that two commuting elements are powers of some element: if $x,y$ are in a free group, then $\langle x,y\rangle$ is free, being subgroup of a free group, with rank ...
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52 views

Proof of a simple isomorphism

I'm having trouble with part b. I know I have to show that a map from a to da is an isomorphism from Z to dZ, but I don't know the general way to prove that a map is an isomorphism. I think I'm ...
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100 views

If $G$ be a cyclic group of prime order $p$, prove that every non-identity element of $G$ is a generator of $G$.

If $G$ be a cyclic group of prime order $p$, prove that every non-identity element of $G$ is a generator of $G$. WE know that no of generator= $\phi{(p)}=p-1$. We have, if $a$ is a generator of ...
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39 views

Writing specific word as product of commutators in free groups

Let $F$ be a free group on $\{a,b,c,\cdots\}$. Then a word $$a^mb^nc^k\cdots \hskip1cm \mbox{(finite expression)}$$ lies in commutator subgroup $[F,F]$ if and only if sum of powers of $a$ is zero, ...
2
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1answer
51 views

Show that if $p$ is a prime number, and $G$ is a transitive subgroup of $S_p$, then $G$ must contain a cycle of length $p$.

Note that a subgroup $G$ of $S_n$ is called a transitive subgroup of $S_n$ if it acts transitively on the set $\{1,2,...n\}$. I understand that, for any $x$, $y$ $\in \{1,2,...p\}$, there exists an ...
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graph automorphism translating each finite set

Consider the following well-known definition: An automorphism $g$ (action on vertices) of a connected (not necessarily locally finite) infinite graph is called translation if $g \cdot F \neq F$ for ...
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63 views

Two groups are isomorphic imply their group algebras are isomorphic?

Is this statement true? Let $G$ and $H$ be a group, and $G \cong H$. Then $\mathbb C G \cong \mathbb C H$? For example I want to show that $\mathbb C G \cong \mathbb C H$ where ...
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80 views

Calculation in a Group Ring

I have some problems with the verification of the third equation in Lemma 1 in this paper. First of all, one has to notice that there is at least one Error in the Definition of $a_{\kappa,\nu}$ ...
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1answer
48 views

Extension of a Set to Basis of Free Group

Let $F$ be free group on $\{x,y\}$. Consider the element $x^2$. Does there exists $z\in F$ such that $\{z,x^2\}$ generate whole $F$? I thought in following direction. $x^2$ is not in commutator ...
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422 views

Do subgroup and quotient group define a group?

Does a (normal) subgroup along with its corresponding quotient group define a group completely? Or are there groups with isomorphic normal subgroups and isomorphic corresponding quotient groups which ...
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20 views

Show that $m(g)=0$ if $g$ is generator of $GF(p^n)$ and $m(x)$ is a prime polynomial for the given field.

As the title states I need to prove the above assertion. Any hint on how I go about this. Show that $m(g)=0$ if $g$ is generator of $GF(p^n)$ and $m(x)$ is a prime polynomial for the given field.
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48 views

What is Amalgamation

My question is what exactly is free product of groups with amalgamation? I came across this term in algebraic topology. Is it possible to explain it at a level that an undergraduate with knowledge of ...
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39 views

$\phi: G\to G'$ be a group homomorphism, is it true that $G/\ker\phi\cong G$?

Let $\phi: G\to G'$ be a group homomorphism and $N$ be a normal subgroup of $G$. If $N=\ker\phi$, then is it true that $G/N\cong G$? That is, is it true that $G/N$ is isomorphic to $G$? Furthermore, ...
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1answer
31 views

Subgroups of Groups with Several Characterizations

The Frattini subgroup is defined for arbitrary group in at least two ways: It is the intersection of all maximal subgroups. It is the set of all non-generators of the group (i.e. it contains those ...
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1answer
37 views

Free Generators (Basis) for a Normal Subgroup of a Free Group

Let $F=\langle a,b\rangle$ be a free group of rank $2$. Its commutator subgroup has a nice free-basis: $$[a^m,b^n], \,\,\,\,m,n\in\mathbb{Z}.$$ Instead of $[F,F]$, we consider another simplest normal ...
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1answer
33 views

Show that $G/N$ acts faithfully on $S$ if and only if $N=\ker\phi$

There is this supposed to be a not-so-difficult proof but somehow I just find it a bit hard to connect the dots. Suppose that $G$ acts on a set $S$, and let $\phi$ be the associated homomorphism from ...
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2answers
36 views

Show if $N$ is normal subgroup of $G$ and $H$ is a subgroup of $G$, then $N \cap H$ is normal subgroup of $H$.

Show if $N$ is normal subgroup of $G$ and $H$ is a subgroup of $G$, then $N \cap H$ is normal subgroup of $H$. attempt: Then recall $N \cap H$ is normal if and only if $h(N \cap H) h^{-1} \subset N ...
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23 views

Does the Bockstein commute with maps induced by group homomorphisms?

Let $G$ be a finite group, and let $\sigma:G\to G$ be a homomorphism of groups. There is an induced map in cohomology $\sigma^*:H^*(G,\mathbb{F}_p)\to H^*(G,\mathbb{F}_p)$. We also have the ...
4
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0answers
83 views

Geometric Coset

I am familiar with cosets, but im not sure what to do here (never had to give a "geometric" description of a coset): Consider $\mathbb{R}$ and the subgroup $\mathbb{Z}$. Describe a coset ...
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1answer
19 views

Proving surjectivity of a function between stabilizer groups

I am working on proving that a certain function is bijective. I think I have proven the injectivity aspect however I'm having troubles proving surjectivity. Let $X$ be a set, $i \in X$ and say that ...