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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2
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1answer
567 views

Every minimal normal subgroup of a finite solvable group is elementary abelian

Show that every minimal normal subgroup of a finite solvable group is an elementary abelian $p$-group for some prime $p$. I'm stuck on this one, any idea is appreciated.
4
votes
2answers
70 views

How to prove that $a^{2^{n-2}} \equiv 1 \pmod{2^n}$?

Let $a$ be an odd integer and $n$ an integer such that $n\ge 3$. 1) I want to show that $a^{2^{n-2}} \equiv 1 \pmod{2^n}$ 2) Then I want to show that $(\mathbb Z/{2^n\mathbb Z})^*$, the ...
1
vote
1answer
101 views

About blocks on permutation groups

I have the following situation: $G$ is a permutation group, transitive on $\Omega$, and $B$ is a non-trivial block. (for every $x\in G$, a block $B$ verifies either $B\cdot x=B$ or $B\cdot x\cap ...
3
votes
4answers
246 views

Ways of expressing permutations as products of transpositions

Determine whether the following permutation is even or odd and write it as a product of transpositions in two different ways. $(1527)(3567)(273)$ So far, I have the following: ...
1
vote
0answers
49 views

What is the value of $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G $ are in the same conjugacy class?

I know the value of this summation $\sum_{i=1}^n X_i(g) X_i(h^{-1})$ when $g,h \in G $ are in the different conjugacy class will be zero and I know how to prove it but what about if they are in the ...
0
votes
1answer
81 views

The characters of the irreducible representations of a group

Let $G$ be finite group of order $n$ with $s$ conjugacy classes and let $X_1, . . . ,X_s$ be the characters of the irreducible representations of $G$ over $C$. Prove that the sum $ \sum_{g\in G} X_i(g) ...
0
votes
1answer
33 views

Constructing elements that meet order requirements in finite group

Let $n$ be an integer. Construct a group $G$ containing two elements $a$ and $b$ of order 2 whose product is of order $n$. My attempt at a solution: If we have that $n$ is a number such that $n-1$ ...
4
votes
1answer
125 views

Prove that there is no homomorphism from $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ ont0 $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$

Prove that there is no homomorphism from $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ ont0 $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$. My idea for the proof : Let $\phi$ be such homomorphism. Since Ker $\phi$ ...
3
votes
0answers
105 views

Reflections in Dihedral Group

In Dihedral Groups, what is the meaning of reflection ? A line needs to be specified for a reflection to take place, but, if you specify only one line how will $D_n$ give all the symmetries for a ...
0
votes
1answer
85 views

π-separable group and subnormal series

I want to show that if $G$ has a subnormal series, then $G$ is $\pi $-separable group. It is enough to show that ${N_i} \triangleleft G$ for every ${N_i} \triangleleft {N_{i + 1}}$ in series. But I ...
4
votes
1answer
122 views

Problems on Sylow Theorems

Let $G$ be a finite group, let $p\in\mathbb{N}$ be a prime and let $$(ab)^p=a^pb^p,~~ \forall a,b\in G$$ Prove that $G$ has a unique sylow $p-$subgroup.
0
votes
1answer
79 views

$\pi$-separable group and its homomorphic image

Show that every subgroup and every homomorphic image of a $\pi $-separable group $G$ is also $\pi $-separable. I find a normal series of a subgroup or homomorphic image but I can not show that every ...
6
votes
2answers
110 views

Is there a way to show $\langle x,y,z: xz=zx,yz=zy,xy=yxz,x^4=y^4=z^2=1\rangle$ has order $8$?

The quaternion group has a particular presentation $$ \langle x,y,z: xz=zx,yz=zy,xy=yxz,x^4=y^4=z^2=1\rangle $$ So it must have order $8$, but can you deduce that just from the relations? The ...
5
votes
1answer
152 views

Group structure of an elliptic curve

Let $E$ be an elliptic curve over field $\mathbb{Z}/p\mathbb{Z}$. The curve group $E(\mathbb{Z}/p\mathbb{Z})$ is always a) cyclic or b) direct product of two cyclic groups. First question: How do I ...
1
vote
2answers
145 views

Let $p$ be prime number. In any finite group, the number of elements of order $p$ is multiple of $p-1$.

This is the problem in the book that I want to prove, but it doesn't seem right. For example let's say I have a group of $9$ elements. If this group is non-cyclic then every element (except identity) ...
0
votes
2answers
114 views

Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$

Claim: Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$ May I know if my proof is correct? Thank you. Lemma: $G/Z(G) $ is ...
1
vote
1answer
40 views

Smallest group with a derived series of length 2, 3 and 4

What are the smallest group with a derived series of length 2, 3 and 4?. I know that for n=2 the answer is S3 because that's the smallest metabelian non-abelian group. Could you help me out? Thanks
1
vote
1answer
347 views

Drawing the subgroup lattice of D10

I've been tasked with drawing a subgroup lattice of the dihedral group of order 10. I know from Lagrange's theorem that non-trivial subgroups must have order 2 or 5. Finding the subgroups of order 2 ...
0
votes
2answers
40 views

Some problems in group theory

May I know if my proof/solution is correct? Thank you v. much. 1.) If $G, H$ are finite groups of order $10$ and $21$ respectively, then every homomorphism $f:G \to H$ satisfies $f[G] = \{e_H\}.$ ...
1
vote
1answer
100 views

Four groups of order 20 that are not isomorphic [duplicate]

Give four groups of order 20 that are not isomorphic. I know the integers under addition mod 20 is one group of order 20, but what would three other groups of order 20 that are not isomorphic to ...
1
vote
0answers
23 views

Construction of an isomorphism between certain subgroups of $GL_2(\mathbb{C})$ and $S_4.$

Consider the following matrices $A:=$$ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} ,\ B: =\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ of multiplicative group $GL_2(\mathbb{C})$ and ...
-4
votes
1answer
77 views

If $\alpha$ is cycle How do I prove that $\alpha^k$ is cycle?

$ord(\alpha)=r$, $k=\frac{r+1}{2}$, $r$ is odd How I show that $\alpha^k$ is cycle and $ord(\alpha^k)=r$?? Thank you! I add new impotent detail...
2
votes
1answer
341 views

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$ May I know if my proof is correct? Thank you v. much. Proof: $$|G:(H \cap K)| \leq |G:H||G:K| ...
0
votes
1answer
41 views

Question about cyclic (Renew Question) [duplicate]

At my Question about permutation - (I add new details, now it shold be more clear)(Question about permutation - (I add new details, now it shold be more clear) I explain it wrong , now I try to ...
0
votes
1answer
135 views

Finding the “square root” of a permutation

Suppose $r$ is odd, and ${\rm ord}(\alpha)=r$. ($\alpha,\beta$ are cycles.) Now, $\alpha=(a_1\cdots a_n)$. I need to find $\beta$ that will make $$\alpha = \beta^2$$ How can I show it? What I ...
3
votes
1answer
79 views

How to show that two groups makes $S_n$

I need to show that: $S=\left\{(12),(13),...,(1n)\right\}$ generates $S_n$ $S=\left\{(12),(123\cdots n)\right\}$ generates $S_n$ How do I show that each one of them generates $S_n$? Thank you!
4
votes
2answers
87 views

How do I prove that $a\in (\mathbb{Z}_{p}^{*})^2 \Leftrightarrow a^{\frac{p-1}{2}}\equiv 1 \pmod p$

$p$ is prime number $>2$ and $a$ is a square. $\mathbb{Z}_{p}^{*} $ is a cyclic group. I need to show that $$ a\in (\mathbb{Z}_{p}^{*})^2 \iff a^{\frac{p-1}{2}}\equiv 1 \pmod p $$ Any ideas ...
11
votes
3answers
409 views

Could the concept of “finite free groups” be possible?

Is it possible to define "finite free groups" ? could that make it easier to deal with group presentations ?
1
vote
0answers
105 views

Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$.

I have read somewhere (do know where) the following statement. Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$. I want to prove it. If there is a ...
9
votes
1answer
168 views

On conjugacy class size of finite groups.

Suppose $G$ is a finite group such that the set of all the conjugacy class size is $\{1,2,\dots,n\}$, where $n$ is a natural number. Is it true that $n\leq 3$? Thanks in advance.
1
vote
2answers
48 views

show H is a subgroup

Let G be a finite abelian group. Let $H=\langle a,b\rangle = \{a^{i}b^{j}\;\;\;i,j\in\mathbb{Z}\}$ Show that $H$ is a subgroup of $G$ My solution. 1) Need to show $H\neq\emptyset$ This is true as ...
1
vote
1answer
101 views

Normal subgroup in center of the group

Let $G$ be a group of order $3825$. Prove that if $H$ is a normal subgroup of order $17$ in $G$ then $H\leq Z(G)$. In the link below, the solution basically says that the index of $C_{G}(H)$ ...
10
votes
3answers
633 views

Find four groups of order 20 not isomorphic to each other.

Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic. So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral ...
3
votes
1answer
144 views

Monstrous Moonshine for $M_{24}$?

This is connected to my MO post "Monstrous Moonshine for $M_{24}$ and K3?". In page 44 of this paper, eqn(7.16) and (7.19) yield, ...
7
votes
0answers
102 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
2
votes
2answers
148 views

Counterexample of Sylow subgroups of a subgroup

Let $P$ be a Sylow subgroup of a finite group $G$. Let $N$ be a subgroup of $G$. (1) If $N$ is normal in $G$, then $P\cap N$ is a Sylow subgroup of $N$. I have proved this. (2) In general, is ...
1
vote
1answer
176 views

Operation table for the quotient group $S_4/V$

So I need to write out an operation table for the quotient group $S_4/V$, where $$V = \{(e),(12)(34),(13)(24),(14)(23)\}$$ thus $|V|$ = 4 and $|S_4|$ = 24. My question is: Do I need to really write ...
2
votes
5answers
148 views

$H<K<G$ and $gHg^{-1}< K$ implies $g^{-1}Hg < K$

Does anyone know of a counter example or a proof of the following proposition? If it doesn't hold in general are there any classes of groups for which it holds? Let $G$ be a non-abelian finite ...
1
vote
2answers
145 views

Prove that $PSL(2,9)\cong A_6.$

I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995). The ...
2
votes
1answer
81 views

Step in Proof of Cardinality of Product of two Groups

Let $A,B < G$ be two subgroups of some group $G$, then I have a question on the proof of the following: $$ |AB| = \frac{|A||B|}{|A \cap B|}. $$ Proof: Let $D = A \cap B$, arrange $A$ and $B$ in ...
0
votes
1answer
332 views

If $H$ is a subgroup of $G$ of finite index $n$, then under what condition $g^n\in H$ for all $g\in G$

A consequence of Lagrange's theorem in finite group theory is that $g^{|G|}=e$ for all $g\in G$. I wonder whether this be generalized along these lines: If $H$ is a subgroup of $G$ of finite ...
1
vote
0answers
106 views

Orders of Elements in Minimal Generating sets of Abelian p-Groups

I'm looking for as much information about the orders of elements in minimal generating sets of finite abelian $p$-groups as possible. What I really need is complete knowledge about the possible orders ...
3
votes
0answers
62 views

An example where $Z(Z_G(A))$ is not a subset of right Engel elements in a finite $p$-group

Find a counter example to the following statement: Let $G$ be a finite $p$-group such that $G/Z(G)$ has exponent $p$. Let $A$ be a normal abelian maximal subgroup of $G$, and $Z_G(A)$ be the ...
0
votes
5answers
120 views

Contemporary Abstract Algebra (External Direct Products):

My question is from the book (Contemporary Abstract Algebra) in chapter 8 exercise 8: Prove that $S_4$ is not isomorphic to $D_4 \times Z_3$. If anyone if may can help me with this problem I ...
0
votes
1answer
67 views

How to prove or disprove finiteness?

How to prove or disprove that statement: a group is finite if the set of all its subgroups is finite?
1
vote
1answer
41 views

Is there a metabelian group satisfying the following conditions

Is there a group $G$ satisfying the following conditions : (a) $G$ is metabelian of class $p$, whose commutator subgroup has exponent $p$; (b) $G$ has no abelian subgroup of index $p$; (c) There is ...
1
vote
1answer
131 views

If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.

I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it ...
7
votes
1answer
97 views

Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
2
votes
1answer
79 views

When is the action of $G$ on $\text{Syl}_p(G)$ by conjugation is double transitive?

We know that the action of $G$ on $\text{Syl}_p(G)$ by conjugation is transitive. I wonder when this action can be double transitive on $\text{Syl}_p(G)$. Thanks for your help.
14
votes
0answers
265 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...