Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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1answer
51 views

Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$

Find the number of subgroups of $\mathbb{Z}/(5)\times \mathbb{Z}/(5)$ including trivial subgroups. My Work: If we consider $\mathbb{Z}/(5)$ then the only subgroups are trivial subgroups. But how ...
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0answers
31 views

How we can conclude that $p\nmid \sum_{x\in H}|x^G|$ in a group with some elements of order $2p$?

Let $G$ be a finite group such that has some elements of order $2p$, where $p$ is an odd prime. Let $H$ be the set of all elements of order $2p$ in $G$. We can show $G$ acts on $H$ by conjugation. So ...
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2answers
93 views

Are the primitive groups linearly primitive?

A transitive permutation group $G \subset S_n$ is primitive if $G_1 \subset G$ is a maximal subgroup. A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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1answer
106 views

Subgroups of Semidirect Product of the elementary abelian group of order 8 by $S_3$

What are the subgroups of the semidirect product of the elementary abelian group of order 8 by $S_3$? This is the group $(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2)\rtimes S_3$ of order 48; ...
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1answer
21 views

Which inclusions of finite groups are relatively linearly primitive?

This post is a sequel of: Which finite groups have faithful complex irreducible representations? A finite group $G$ is linearly primitive if it has a faithful complex irreducible representation. ...
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2answers
211 views

A group of order $pqr$ (primes $p > q > r$) has a subgroup of order $qr$ [duplicate]

I've done most of the following problem, but I can't seem to get part (d). Let $G$ be a group of order $pqr$ for primes $p > q > r$. By a counting argument one can see that there is ...
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1answer
48 views

$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$

Let $G$ be a characteristically simple finite group, i.e. it has no nontrivial characteristic subgroups. Prove there is some simple group $T$ such that $G \cong T \times T \times \cdots \times T$. ...
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1answer
170 views

The finite groups with an irreducible faithful complex representation

All the groups below are supposed finite, and their representations, complex. An abelian group admits an irreducible faithful representation iff it is cyclic. A group has all its non-trivial ...
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0answers
52 views

What is the most mundane & intuitive meaning of the radical of a finite group?

I am asking this question because I am trying to solve this problem from a class note: Assume that $R(G)$ is simple and not commutative, show that $G$ is a subgroup of $Aut(R(G)).$ The note's ...
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1answer
31 views

Is comaximal equivalent to simple?

A finite group $G$ is called comaximal if for any non-trival irreducible representations $V$ and $W$ of $G$, it exists $n \in \mathbb{N} \ $ such that $(V^{\otimes n},W)\ge 1$. A finite group $G$ is ...
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2answers
74 views

Group isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ [closed]

Is a group with $9$ elements such that all elements (excepted the natural element) are of order $3$ is isomorphic to $\Bbb Z/3\Bbb Z\times \Bbb Z/3\Bbb Z$ ?
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0answers
111 views

Is there a transitive permutation group satisfying these properties?

Let $G \subset S_n$ be a transitive permutation group and let $H=G_1:=\{ g \in G \ \vert \ g(1)=1 \}$. Let $(K_i)_{i \in I}$ be the sequence of minimal overgroups of $H$ in $G$. Note that if $G$ is ...
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1answer
54 views

Does the number of elements of order $r$ equal $\sum_{|x| = r} |x^G|$?

I want to know if this sentence is true or not? Let $m_r$ be the number of elements of order $r$ in a finite group $G$ and also let $x$ be an element of order $r$ in this group, then ...
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1answer
54 views

Finitely generated abelian groups isomorphism

Got this on a home assignment and I don't have a clue... How do I determine if $\mathbb{Z}_{12}\times\mathbb{Z}_{18}$ and $\mathbb{Z}_{6}\times\mathbb{Z}_{36}$ are isomorphic? Any hints will be very ...
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0answers
182 views

A property of the subgroups lattices

Let $G$ be a finite group. Consider all the subgroups $H$ such that its subgroups lattice $\mathcal{L}(H)$ is distributive (i.e. the group $H$ is cyclic, by Ore's theorem), and among them, let $\{ ...
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1answer
42 views

stabilizers and orbits of elements in $S_4$

Let $S_4$ be the group of symmetries of $4$ letters, and let $S_4$ act on itself by conjugation. Let $\sigma = (12)$, $\tau = (123)$, and $\rho = (1234)$. Find the stabilizers of the elements ...
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1answer
260 views

Relation between irreducible and completely reducible representations

While studying representations of finite groups I got confused by the the statement that any irreducible representation is at the same time a completely reducible representation. This doesn't seem to ...
0
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1answer
52 views

$(\mathbb Z/6\mathbb Z)/(2\mathbb Z/6\mathbb Z)$

$$(\mathbb Z/6\mathbb Z)/(2\mathbb Z/6\mathbb Z)=\left\{ \left\{ 6\mathbb Z,2 + 6\mathbb Z, 4 + 6\mathbb Z\right\} ,\left\{ 1+6\mathbb Z,3 + 6\mathbb Z, 5 + 6\mathbb Z\right\} \right\} $$ Is that ...
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0answers
210 views

Constructing irreducible representations of quaternion group over $\mathbb{Q}$

I am a beginner in studying the representation theory and I am doing some exercises in this field. So this is not a homework. My question is about constructing all irreducible representations of ...
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1answer
91 views

Characteristic subgroups of a direct product of groups

Let $G=H\times K$ and $H\times 1$ be a characteristic subgroup of $G$. Then can we conclude that $1\times K$ is also a characteristic subgroup of $G$? My motivation is the case where orders of ...
2
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1answer
45 views

On finite $p$ -group of class two with cyclic center

Do there exist a finite $p$-group of class two such that $Z(G)$ is not a subgroup of $\Phi(G)$, where $Z(G)$ is center of $G$ and $\Phi(G)$ is frattini subgroup of $G$. $\dfrac{G}{Z(G)}$ is ...
1
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1answer
54 views

Number of k-products of disjoint cycles in the symmetric group S(n)

Suppose that S(n) denotes the group of all permutations of the set {1,2,...,n} with the usual composition operation. Is there any formula or expression for n(k), where n(k) denotes the number of ...
0
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1answer
57 views

How many elements $a$ with $o(a)\mid 5$ in $\mathbb{Z}/360\mathbb{Z}$

I am trying to understand why there are exactly $5$ elements of order dividing $5$ in $\mathbb{Z}/360\mathbb{Z}$.
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2answers
149 views

an abstract algebra question: how to prove there exists a subgroup whose order is K

If k is an odd and positive integer, please prove that any group whose order is 2k has a subgroup whose order is k. I have been struggling for this, I tried to use mathematical induction to prove it, ...
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2answers
223 views

Normal subgroup of group order 8

Prove that group of order 8 has hormal subgroup of order 4. I know how to prove that if $G$ has a prime power order $p^n$ then it has a subgroup of order $p^m$ $\forall m \in \mathbb N: 0 \le m \le ...
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1answer
137 views

John Beachy's Sylow Problem & Solution

I am reading John Beachy's online Abstract Algebra which has this question and its solution: PROBLEM: Prove that if $G$ is a group of order 56, then $G$ has a normal Sylow 2-subgroup or a normal ...
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0answers
82 views

proof theorem 13.9 on finite permutation groups of Wielandt book

I can't prove this theorem on finite permutation groups of Wielandt book. Theorem 13.9: Let p be a prime and G a primitive group of degree n=p+k with k≥3. if G contains an element of degree and order ...
2
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1answer
56 views

Is the solution of this trivial group theory exercise correct?

If $G$ is a finite group, prove that, given $ a \in G $, there is a positive integer $n$, depending on $a$, such that $a^n=\bf1$, where $\bf1$ is the identity element of the group. This is my proof: ...
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1answer
59 views

Non-abelian rank of a finite abelian group

By rank $rk\ G$ of a group I mean the minimum set of its generators (even if the group is abelian, I consider it as if it were just a general group $-$ I do not mean here torsion-free rank!). For a ...
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1answer
101 views

Subnormal $p$-Subgroup, Layer, Fitting & Radical Subgroups

I am self-studying a class note on finite group and come across a problem like this: Let $G$ be a dihedral group of order 30. Determine $O_2(G),O_3(G),O_5(G), E(G),F(G)$ and $R(G).$ Where ...
3
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1answer
99 views

Presentation of a non-abelian group of order $p^4$ such that ${G}/{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$

Let $G$ be a finite non-abelian $p$-group of order $p^4$ and $\frac{G}{\Phi(G)}\cong \Bbb{Z}_p\times \Bbb{Z}_p$, where $p$ is a prime. What is the presentation(s) of $G$?(If $G$ exixsts). Thanks ...
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1answer
73 views

Did I map this group right?

I am trying to get a group with 8 elements. This is a Cayley table of $\mathbb Z_2 \times \mathbb Z_4$. Is this right? $$ \begin{array}{c|cccc} & 1 & 2 & 3 & 4\\ \hline 1 & 1 ...
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1answer
39 views

Subgroups isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5} $

Let $A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}$. I need to calculate the number of subgroups of $A$ which is isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$ ...
2
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1answer
65 views

$P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times Q$.

suppose that $G$ is finite group and $P$ is aabelian $p$-sylow subgroup of $G$ and $H=N_{G}(P)$. show that $P$ is normal complement in $G$ if and only if $H$ has normal subgroup $Q$ which $H=P\times ...
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0answers
126 views

Galois group of $x^8-2$ and intermediate fields

Let $p(x)=x^8-2$ be a polynomial over $\mathbb{Q}$. I am to find its Galois group $G$ and the correspondence between subfields of its splitting field $\mathbb{K}_p$ and subgroups of $G$. It is easy to ...
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1answer
84 views

How many elements of order 5 may contain in a group of order 90? [closed]

G - group of order 90 THE QUESTION: How to found the count of elements of order 5 in this group?
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1answer
58 views

how to show that $P$ is normal complement in $H$.

Suppose that $G$ is a finite group and not simple,and $H$ is a minimal normal subgroup of $G$,and also $P$ is a Sylow $p$-subgroup of $H$. if $P$ is abelian and $\frac{N_{G}(P)}{P}$ will be a group ...
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2answers
54 views

$G$ contains at least $r(p-1)$ elements of order $p$

Suppose a group $G$ has $r$ distinct subgroups of prime order $p$. Show that $G$ contains at least $r(p-1)$ elements of order $p$. Aside: I know how to use this to prove that a group of order $56$ ...
3
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1answer
134 views

HK is cyclic if H and K are both cyclic

Let $G$ be a finite group with normal subgroups $H$ and $K$ of relatively prime orders. Show that the group $HK$ is cyclic if $H$ and $K$ are both cyclic. My attempt was to use the $2$nd Isomorphism ...
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371 views

Group of Order 33 is Always Cyclic

I would love to get help on this problem from a chapter on Commutator of Group Theory: Show that each group of order 33 is cyclic. (Hint: Use the result from the Exercise and Lemma below.) ...
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38 views

$\mathrm{Aut}(G)$ vs. $\mathrm{Aut}(H)$ where $H$ is a maximal abelian subgroup

Can I find a finite group $G$ and a maximal abelian subgroup $H$ such that $ \mathrm{Aut} (H)$ is not isomorphic to a subgroup of $\mathrm{Aut}(G)$?
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4answers
96 views

If $G$ is a finite group and $H$ is a subgroup of $G$, then $|H| = |gH|$ for every $g \in G$.

I'm stuck. I believe I have half of the proof, but I'm missing an essential part to complete the proof. Any hints would be appreciated. Thanks! Proof: Suppose $G$ is a finite group and $H$ is a ...
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2answers
118 views

Chevron Symbols in Commutator

I am reading a chapter on Commutator in Group Theory and came across chevron symbols "$\langle$" and "$\rangle$" like these: Question #1: Let $E$ and $F$ be non-empty subsets of $G$, we set ...
4
votes
1answer
84 views

Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$

Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times ...
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2answers
50 views

herstein excercise on a finite group

I'm stuck on this herstein exercise for a long time. Let $P$ is a $p$-Sylow subgroup of $G$ and order of $a$ is a prime power then if $a\in N(P)$ prove $a\in P$ I was doing like this but stuck in ...
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0answers
62 views

Over which fields is a $G$-module reducible?

Let $K$ be a field of characteristic zero, or if this is too general, an algebraic number field. Let $G$ be a finite group and $V$ an irreducible and finite-dimensional $KG$-module. Let $\chi$ be the ...
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2answers
809 views

Group of order 30 can't be simple

I have this following question from my class note on Sylow Theorem: Show that a group of order 30 can not be simple. For that I know the followings: (1) A simple group is one that does not have ...
0
votes
1answer
48 views

Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$

I have this following question from my class note on Sylow Theorem: Let $p$ and $q$ be prime numbers such that $p \nmid (q-1).$ Show that each group of order $pq$ possesses a normal subgroup of ...
6
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1answer
145 views

A question about the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$

I wanted to clarify some confusion I was having on the automorphism group of $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$, which I call $Aut(\mathbb{Z}_{2} \times \mathbb{Z}_{4})$. I considered the ...
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1answer
54 views

What does “exponent 2 nilpotency class 2” mean?

According to the book The Symmetries of Things, p. 208, the number of groups of order 2048 "strictly exceeds 1,774,274,116,992,170, which is the exact number of groups of order 2048 that have ...