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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143 views

Finite group with the property that every element of order a power of $p$ is contained in a conjugacy class of size a power of $p$

Let $p$ be a prime and let $G$ be a finite group with the property (*) that every element of order a power of $p$ is contained in a conjugacy class of size a power of $p$. i) If ...
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3answers
52 views

irrep of a non unit element in the finite group

Let $G$ be a finite group. Prove following statement. $\forall g \in G$ such that $g \neq e$ $ \exists$ a complex irrep $\rho$ such that $\rho(g) \neq E$ I have no idea how to start it. I can prove ...
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1answer
55 views

Factoring over prime fields

Suppose I have two numbers in Fp that are multiplied together: r*s Is there anything special that needs to be done when prime factorizing since this is a finite field (i.e., is this what is referred ...
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1answer
73 views

Finite group that has order not divisible by 3

If $o(G)=m$ and $m$ is not divisible by $3$ then show that for every element $x\in G$ there exists another element $y\in G$ such that $x = y^3$.
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1answer
42 views

number of complex irreps

Can irreducible complex representation of a finite group of be exhausted by a) two 1-dimensional and two fifth-dimensional representations? b) five one-dimensional and 1 five-dimensional? My ...
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1answer
114 views

Nonabelian group of order $p^4$ [closed]

Let $P$ be a nonabelian group of order $p^4$, where $p$ is a prime, and let $A$ be a subgroup of $P$ maximal with the property of being normal and abelian. Prove that $A$ is of order $p^3$. Thanks a ...
1
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1answer
38 views

Qustion about field and sub-group

$F$ is a field and and $H$ is finite sub-group of $(F,\cdot)$ ($F$ without the $0_F$). I need to prove that $H$ is cyclic. I can use this fact - Can we conclude that this group is cyclic?. (I don't ...
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2answers
153 views

A finite group with the property that all of its proper subgroups are abelian

Let $G$ be a finite group with the property that all of its proper subgroups are abelian. Let $N$ be a normal subgroup of $G$. Prove that either $N$ is contained in the center of $G$ or else $G$ has a ...
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1answer
236 views

Group of order $1575$ having a normal sylow $3$ subgroup is abelian.

Question is to prove that : If a group $G$ with $|G|=1575=3^2\cdot5^2\cdot 7$ has a normal sylow $3$ subgroup then : sylow $5$ subgroup is normal sylow $7$ subgroup is normal In this ...
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2answers
61 views

How to apply Thompson's A×B lemma to show this nice feature of characteristic p groups?

How does one apply Thompson's A×B lemma (Lemma 24.2 on page 112 of Aschbacher's Finite Group Theory) in order to prove this nice lemma (Lemma 31.16 on page 160)? In the book, I basically don't ...
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1answer
150 views

Use of the commutator to reduce the degree of a permutation

I've found the following claim here (page 4, in the proof of thm 1.1). Set $A$ the support of a permutation $\tau$ in $S_n$, with $\deg\tau=|A|=k$. Given a permutation $\pi$ in $S_n$ such that ...
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1answer
68 views

Cohomology of finite p-groups

Given a finite abelian $p$-group $A$ acted on by a finite $p$-group $G$. Under the assumption $\operatorname{H}^1(G,A_1)=0$, where $A_1$ is the set of elements of $A$ having order at most $p$, what ...
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4answers
164 views

Semi-direct product of groups with one of them cyclic

Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove ...
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2answers
254 views

surjective homomorphisms between cyclic groups (wrong question)

This statement does not hold. Let $C, D$ be cyclic groups and $f_1,f_2:C\rightarrow D$ be surjective homomorphisms. Show that there exists $\xi: C\rightarrow C$ such that $f_2=f_1\circ \xi$. My ...
3
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1answer
110 views

Let $G$ be a group of order $360$ and suppose $M$ is a maximal subgroup of $G$ which is isomorphic to $A_5$. Prove that $G$ is isomorphic to $A_6$.

Let $G$ be a group of order $360$ and suppose $M$ is a maximal subgroup of $G$ which is isomorphic to $A_5$. Prove that $G$ is isomorphic to $A_6$. Help me some hints. Thanks a lot.
4
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1answer
128 views

Prove that $G=S_{1}\times S_{2}\times\cdots\times S_{k} $

Let $G$ be a finite group and let $N_1,N_2,\ldots,N_k$ be normal subgroups. Suppose that $\bigcap_{i=1}^{k}N_{i}=\{1\}$ and that $G/N_i=S_i$ is simple. If all the groups $S_i$ are non-isomorphic ...
3
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1answer
48 views

Showing all p-local subgroups are char p given that all are contained in char p locals

Question: How does one prove that if every $p$-subgroup $U$ is contained as a subnormal subgroup of a characteristic-$p$, radical, local subgroup containing the normalizer of $U$, then the normalizer ...
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1answer
64 views

Prove there is an element $x \, \epsilon \,G$ with $\left |x \right | = 2$ if and only if $\left |G \right |$ is even

1) Let $G$ be a finite group. By considering the size of the set $ \left \{ x \epsilon \, G \, : \left |x \right | \geqslant 3 \right \}$ prove there is an element $x \epsilon G$ with $\left |x ...
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1answer
233 views

Abelian subgroup in the symmetric group

Let $p$ be a prime number. Show that there is an abelian subgroup $P$ of order $p^p$ in $S_{p^2}$ such that every element in $S_{p^2}$ that isn't in $P$ does not commute with every element in $P$. ...
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1answer
90 views

Show $S_4$ is not isomorphic to $D_{24}$ by looking at their centers

Every proof seems to use an argument that looks at the orders and finds an element with a certain order in $S_4$ and no element in $D_{24}$ has that order... Wouldn't it be much much easier to look ...
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1answer
88 views

semidirect product problem

I am trying to show that the semidirect product of $G$ with $G$, where $G$ is a finite group, with the automorphism by conjugation on itself, is isomorphic to direct product of $G$ with $G$. Please ...
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1answer
45 views

Group theory: group actions on finite group.

I'm having trouble with the following question: Let $G$ be a finite group acting on a finite set $X$. For $g\in G$, let $Fix_X(g) =\{x\in X \mid xg = x\}$ and, for $x\in X$, let $G_x = \{g\in G \mid ...
4
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2answers
180 views

Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
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2answers
100 views

Suppose a unique a generates a cyclic subgroup of order. Show ax = xa. - Fraleigh p. 67 6.50

(1.) I don't understand above. How do you magically envisage and envision to let $b = xax^{-1}$? What I did was to start from the answer and see if I can get a chain of equivalences. $ax = xa ...
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4answers
401 views

Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.

This solution is from here and yahoo. Given $a,b$ elements of $G$, and $ab$ has finite order $n$. Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$. Need to show $n$ is the smallest positive integer ...
3
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1answer
61 views

An element of order 5 whose powers are conjugate

Find a finite group G such that it contains an element g of order 5 and its power $g, g^2, g^3, g^4$ are conjugate elements.
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2answers
89 views

Finite abelian groups of odd order

I am reading this paper. It is about finite abelian groups of odd order. I need to find maximal subset which doesn't contain 3-term arithmetic progression. I don't understand the need of odd order. I ...
3
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2answers
161 views

$x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$ [duplicate]

Problem Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $\vert xy \vert = n$. Solution We have $x^2 = ...
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2answers
70 views

Order of a cyclic group with a single proper subgroup of order 7

Let $G$ be a cyclic group with its only proper subgroup of order 7. Find out the order of the group. let the subgroup of order 7 be denoted by H. since 7 is prime H is cyclic. Now if G = then ...
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2answers
90 views

Isomorphism problem involving the Symmetric Group [duplicate]

Let $G$ be a group of order $24$ with no elements of order $6$. Prove that $G$ isomorphic to the symmetric group $S_4$. This is what I did and I'm a bit unease about it since I didn't use that G ...
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2answers
56 views

Relationship between $|N_G(K)|$ and $|N_H(K)|$ for $K\le H\le G$

Suppose $K\le H\le G$ are finite groups. I'd like to know when the following equation holds: $$|N_G(K)|=|N_H(K)|\cdot [G:H]$$ A sufficient condition is the normality of $K$ in $G$. In general, I'm ...
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1answer
114 views

prove that the Galois group $Gal(L:K)$ is cyclic

Let $L$ be a field extension of a field $K$. Suppose $L=K(a)$, and $a^n\in K$ for some integer $n$. If $L$ is Galois extension of $K$ (i.e., $L$ is the splitting field of $f(x)\in K[x]$, and $f$ is ...
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0answers
21 views

isometry group of an integer $n$ as of the corresponding $\Omega(n)$-parallelotope

Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal ...
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0answers
97 views

Let $ G$ be $SL_2(\mathbb{F}_5)$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$.

Let $G$ be $SL_2(\mathbb{F}_5)$ i.e. the special linear group of $2\times 2$ matrices $\mathbb{F}_5$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$. I ...
7
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1answer
238 views

An automorphism of a finite group which sends more than three quarters of elements to their inverses [duplicate]

Question is : Let $G$ be a finite group and suppose the automorphism $T$ sends more than three quarters of elements of $G$ onto their inverses. Prove that $T(x)=x^{-1}$ for all $x\in G$ and that $G$ ...
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3answers
212 views

Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
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3answers
123 views

Showing H is a normal subgroup of G [duplicate]

Show that if $G$ is a finite group of order $n$, and $H$ is a subgroup of order $\frac{n}{2}$, then $H$ is a normal subgroup of $G$. Please help me on this. I only know that $|gH| = |H|$. How ...
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2answers
73 views

Counterexample that $a\in G$, $a^n\notin H$, for $H$ a subgroup of finite index $n$ in $G$. [duplicate]

Let $G$ be a group and $H$ a subgroup of finite index $n$. Give a counterexample that $a\in G$, $a^n\notin H$ (although I can prove that there exists $k\in\{1,2,\dots,n\}$ such that $a^k\in H$). ...
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2answers
223 views

finite group whose only automorphism is identity map

Question is to prove that : A finite group whose only automorphism is identity map must have order at most $2$. What i have tried is : As any automorphism is trivial, so would be inner ...
2
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1answer
672 views

Finding all homomorphisms between two groups - couple of questions

Consider $\mathbb{Z}_{15}$, and $\mathbb{Z}_{18}$. Let's say I want to find all homomorphisms $f:\mathbb{Z}_{15}\rightarrow \mathbb{Z}_{18}$. I'm not interested in the answer in particular, mostly ...
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1answer
44 views

The dimension of space of morhisms as the number of orbits

All groups are finite, all representations are over $\mathbb{C}$ (just in this context, of course), $G$ is a group, $K,H\subset G$ - its subgroups. By $\mathbb{C}$ we denote the trivial representation ...
2
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2answers
90 views

About the construction of semidirect products

I need help with the following question: We have $G_0=H_0[N]_\rho$ a semidirect product, and $\pi:H\to H_0$ an epimorphism such that $K=\ker(\pi)\unlhd H$, and $H/K\cong H_0$. We have to construct a ...
5
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1answer
146 views

Action on G via Automorphism

Here is an exercise from Isaacs, Finite Group Theory, $4D.1$: Let $A$ act on $G$ via automorphism, and assume that $N \trianglelefteq G$ admits $A$ and that $N \geq C_G(N)$. Assume that ...
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1answer
124 views

An exercise about p-solvable

I'm dealing with a problem about p-solvable in Isaac's finite group theory book. Question is the following: "Let $G$ be $p-$solvable and $P \in Sy{l_p}\left( G \right)$ and $K \le G$ such that $p$ ...
3
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6answers
359 views

Any two groups of three elements are isomorphic - Fraleigh p. 47 4.25(b)

The answer has no details. Hence maybe the answer is supposed to be quick. But I can't see it? Hence I took two groups. Call them $G_1 = \{a, b, c\}, G_2 = \{d, e, f\}$. Then because every group has ...
0
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2answers
105 views

Subgroups of a group

If we have a group $G$ of order $10$, and $H$ is a subgroup of $G$ then Lagrange's theorem states each $H$ may be of order $1, 2, 5$ or $10$. Now we have the trivial subgroup $\lbrace e\rbrace$ of ...
4
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2answers
115 views

When does $x^{n}=y^{n}$ imply $x=y$.

Let $G$ be a finite group of order $m$. Let $n$ be relatively prime to $m$. Let $x,y\in G$ such that $x^{n}=y^{n}$ prove that $x=y$. I was able to prove this result if $G$ is abelian using the map ...
2
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0answers
56 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
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4answers
71 views

Order of elements within a group

If $G$ is a finite group of order (size) $n$ then, for any $g \in G$, the order (period) of $g$ is a divisor of $n$. Proof: $g$ must have finite order since $G$ is finite. If the order (period) of ...
3
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2answers
91 views

Conjugation and generators

Let $G$ be a finite group and $x\in G$ be of order $4$. So $o(x)=4$. Suppose that all cyclic subgroups of $G$ of order 4 are conjugate. Show that there exists an involution $g\in G$ such that ...