# Tagged Questions

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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### If $G = S_5$ and $g = (1 2 3)$, what is the number of elements in $H = \{x \in G \ :\ xg = gx\}$?

Where $H$ is a subgroup of $G$, how could I find $|H|$?
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### A regarding of state vector

A state vector X for a four-state Markov chain is such that the system is four times as likely to be in state 3 as in 4, is not in state 2, and is in state 1 with probability 0.2. Find the state ...
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### How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
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### A question about normal subgroups of nilpotent group

Assume G is a nilpotent group and to any n dividing $|G|$, if there is always a normal subgroup of G with order n?
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### Show that $f$ is a homomorphism.

There is a group $G$ of order $p^3$, where $p>2$. Show that $f:G\rightarrow Z(G)$ with $f(x)=x^p$ is a homomorphism. My attempt: Case a): Suppose $|Z(G)|=p^3$. Then $G=Z(G)$, so $G$ is abelian, ...
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### topic between algebra and geometry [closed]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
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### Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
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### Let $O_{p^{\prime}}(G/A) = T/A$, Why $T \leq F$ and $[A , T]=?$

Let $G$ be a soluble group and $A$ be a minimal normal subgroup of $G$,where $A$ is an elementary abelian group of prime power order. Let each chief factor of $G/A$ has order $4$ or a ...
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### existence of a subgroup of a solvable group G of order d for any divisor d of |G|, with d<|G|^(1/2)

Let $G$ be a solvable group of order $n$ and $d<\sqrt{n}$ be any divisor of $n$. Is there any subgroup of $G$ of order $d$?
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### Let $F = VN$ that $V \cap N = 1$ . Let $L = N_{G}(V)$. $(\vert N \vert , \vert F/N \vert) =$?

Let $G$ be a soluble group and $A$ be a minimal normal subgroup of $G$,where $A$ is an elementary abelian group of prime power order. Let each chief factor of $G/A$ has order $4$ or a ...
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### Cubic Planar Graphs have $2^m-1$ Hamilton Cycles, contradicting Bosak…

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
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### Let N = Fit(G). Why $N = O_{p}(G)$ and $A \leq Z(N)$?

Let $G$ be a soluble group and $A$ be a minimal normal subgroup of $G$,where $A$ is an elementary abelian group of prime power order. Let each chief factor of $G/A$ has order $4$ or a ...
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### Presentations of the unity group

I have been told Bernhard Neumann wrote an article on how to concoct presentations of the trivial group $G=\{1_G\}$. I was curious to see examples of presentations of this simple group. I googled for ...
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### Prove that if $H$ is a unique subgroup of order $|H|$ then $H$ is a characteristic subgroup.

Let $G$ be a group and $H\leq G$. $H$ is a unique subrgroup of order $|H|$. Prove that $H$ is a characteristic subgroup. I tried to show by contradiction that if $\phi (h)\not \in H$ then I can find ...
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### Why does the Principle of Well-Ordering imply a remainder of $0$ for the division algorithm?

I'm currently reading a text (Thomas W. Judson, Abstract Algebra - Theory and Applications) where the author proofs the theorem that every subgroup of a cyclic group is cyclic. The proof goes as ...
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### Finding a normal and not normal subgroup of $S_3$

I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier. I would do this by trial and error, but since the ...
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### Homomorphism of Free Groups

I am reading the theorem of homomorphism of free group from Fraleigh's text in $\S$36 and could only get a fuzzy idea at best: Let $G$ be generated by $A = \{a_i \mid i \in I \}$ and let $G'$ be ...
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### Relations between the Latin squares of order n and the groups of order n.

Given a group $g$ of order $n\in \mathbb{N}$, it is clear that its Cayley table has the Latin square property. That is, every row and column of the Cayley table contains precisely the elements of $g$. ...
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### Is this parabolic induction?

In one of my previous questions, @PL. explained the idea of parabolic induction. In my bachelor thesis I used a technique quite like this, to find all irreducible representations of a certain group. I ...
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### Showing Sylow $p$-groups are Abelian if $n_p=2p+1$

An old qual question on Sylow $p$-groups: Assume that $p$ is an odd prime and $G$ is a finite simple group with exactly $2p+1$ Sylow $p$-groups. Prove that the Sylow $p$-groups of $G$ are abelian. ...
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### What is the purpose of the almost maximal and $p$-supersoluble subgroup?

Suppose that $H \unlhd G$ such that $G/H$ is supersoluble. Suppose that there is an element $y \in H$ such that $H = \langle y \rangle L$ for any almost maximal subgroup L of $H$; then $G$ ...
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### How do we identify $\mathfrak{R}$-automorphisms of a group?

If $G$ is a finite group, a bijection $f\colon G\to G$ is called a (normed) $\boldsymbol{\mathfrak{R}}$-automorphism if $f$ maps subgroups of $G$ to subgroups of $G$, and $f(gH) = f(g) f(H)$ for any ...
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### Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea (...
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### Exercise about finding group isomorphisms

So, i've just learned about groups homomorphisms and isomorphisms. I know that an homormorphism betweet two groups is a function such that $$\phi(a\square b) = \phi(a)\star \phi(b)$$ And when the ...
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### All primes $p$ for which $x^{2} +1$ irreducible over $\mathbb{F}_{p}$

Let $\mathbb{F}_{p}$ be the field with $p$ elements. Determine all primes $p$ for which $x^{2}+1$ is irreducible over $\mathbb{F}_{p}$. Here's what I've got. I think this might be finished but I'm ...
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### Subgroup of a group with 24 elements.

Suppose that $G$ is a finite group of order $24$, which has four $3$-sylow subgroups. We know that may contain $1$ or $3$ 2-sylow subgroup. How can I prove that there only exists one $2$-sylow ...
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### Consequence of First Homomorphism Theorem?

Let $\phi:G\to\bar G$ be a surjective group homomorphism with kernel $N$. Then the first homomorphism theorem tells us that $G/N\cong\bar G$. My question is this: Lagrange's theorem also tells us ...
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### A question on subgroups of a finite group

Let $G$ be a finite group. Let $K$ and $H$ be subgroups of $G$, where $K$ is normal in $G$ and $\gcd([G:H],|K|)=1$. Prove that $K$ is a subgroup of $H$. So far we found that $o(K)$ divides $o(H)$...
A cycle with only two elements is called a transposition. For example, the permutation of $\{1, 2, 3, 4\}$ that sends $1$ to $1$, $2$ to $4$, $3$ to $3$ and $4$ to $2$ is a transposition (specifically,...
### Dihedral subgroups of $S_4$
Prove that in $S_4$ there are $3$ groups that are isomorphic to $D_4$. I know that the $2$-sylows of $S_4$ should be subgroups of order $8$, but to prove it is a bit tricky for me Any help would ...