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
52 views

A detail in Baer Theorem

I refer to Martin Isaac book, "Finite Group Theory", Theorem 2.12 pag. 55. In the proof there is a detail I can't understand. Our hypotesis are the following: $G$ finite group, $H\leq G$ s.t. ...
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0answers
49 views

Finding subgroups via short exact sequences

My professor recently mentioned (when asked how to do a homework problem in office hours) the following technique for finding subgroups of $\mathbb Z \times \mathbb Z$: consider the short exact ...
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1answer
114 views

Is the group $(G,*)$ abelian?

Let $(G,*)$ be a finite group in which the sets $C_a$={$x\epsilon G$|$ax=xa$} have the same cardinality, for all $a \epsilon G$ \{e}. My question is: is the group abelian?
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2answers
107 views

Using the Fundamental Theorem of Finite Abelian Groups

Let $G$ be a finite abelian group and let $p$ be a prime that divides the order of $G$. Use the Fundamental Theorem of Finite Abelian Groups to show that $G$ contains an element of order $p$. The ...
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3answers
189 views

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$? [duplicate]

Let $ (G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
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1answer
55 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
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2answers
185 views

Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
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1answer
56 views

About product of $k$ commutators

I need help in folloving lemma. Lemma: Any product of $k$ commutators is expressible in the form $a_1^{-1}a_2^{-1}...a_{2k}^{-1}a_1a_2...a_{2k}$ I guess author means that any product ...
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1answer
39 views

Specify finite group satisfying two conditions

Let $G = \left\{ 0, 1, 2, 3, 4, 5, 6, 7\right\}$ be a group with the operation $\circ$, which satisfies the following conditions: \begin{align} a \circ b \leq a + b \quad & \forall a,b \in G & ...
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1answer
322 views

If a finite group $|G|$ acts transitively on a set $X$ with $|X|=2^n$, $n \geq 1$, then $G$ has an involution with no fixed points

Let $G$ be a finite group acting transitively on a set $X$, where $|X| = 2^n$ for some $n \geq 1$. Show that some element of $G$ acts as an involution with no fixed points. While it is fairly easy ...
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1answer
59 views

Unique intermediate subgroup and double coset relation II

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is there $\alpha \ge 1$ such that if ...
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2answers
165 views

Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
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1answer
41 views

Find position on number in sorted array

I am trying to calculate some thing and I got lost. I have sorted(low to high) array of $N$ numbers, with first number $K$ and numbers sum of $S$. Assuming that there are no duplicated numbers and ...
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2answers
53 views

Groups and subgroups

I have been told that {0, 2, 4, 6, 8} is a subgroup of the multiplicative integer mod 10. I know that the operation is multiplication, so I understand that every element has its inverse within the ...
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1answer
50 views

Is there a simple group and a proper subgroup with a unique and equidistant intermediate?

Let $G$ be a simple group and $H$ a proper subgroup such that there is a unique intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it possible that $[G:K]=[K:H]$ ? ...
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2answers
60 views

How can I prove that $C_{S_4}((12)(34))$ is a subgroup of order 8?

I would like to know if there is a smart method (i.e. avoid to make all by hand) to understand the order of $C_{S_4}((12)(34))$. The only thing I thought is that ...
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0answers
84 views

Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
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1answer
137 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
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1answer
144 views

Show that the p-Sylow subgroup is normal in $G$

Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x) = x$ implies that $x = e$. Prove that for ...
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1answer
79 views

Prove $G$ has a normal Sylow subgroup

Let $|G|=pqr$ where $p, q$ and $r$ are prime and $p < q < r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$. Let $n_p, n_q, n_r$ denote the number of Sylow subgroups for ...
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0answers
50 views

irreducible polynomial of $\alpha$ over $\mathbb{Q}$

Let $\epsilon=\exp({\frac{2i\pi}{9}})$ and $\mathbb{Q}(\epsilon)$ be a cyclotomic extension of $\mathbb{Q}$ of order 9. If $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{Q}(\epsilon)$, determine the ...
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2answers
63 views

Find all the groups $G$ such that $|G|\leq 6$

Problem statement: Find all the groups of order at most 6. Attempt at a solution: What I thought was, if $|G|=1$, then the only possible element of the group is the neutral element. Now note that ...
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1answer
112 views

Unique intermediate subgroup and double coset relation I

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$). Question: Is it true that $HgK=KgH$, ...
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1answer
63 views

Prove that $G$ is the internal direct product of their normal subgroups $N_1,N_2,\ldots ,N_n$?

Let $G$ be a finite group and $N_1,\ldots,N_n$ are normal subgroups of $G$ such that $G = N_1,\ldots,N_n$ and $o(G) = o(N_1)\cdots o(N_n)$. Then $G$ is the internal direct product of ...
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1answer
42 views

What are all of the possible orders of $K$? Justify your answer.

Let $K\le A_5$. Assume that $K$ is cyclic. What are all of the possible orders of $K$? Justify your answer. So I know that $|K|\in\{1,2,3,5\}$ but I'm not sure how to justify it.
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1answer
32 views

When would a finite group be cosidered as Fp G -Module?

What conditions are necessary to think of a finite group as Fp G -Module ?
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1answer
84 views

Normalizer of a Sylow subgroup

In this question Sylow subgroups of soluble groups Jack Schmidt mentions that the normalizer of $P$ in $S_p$ is solvable. Suppose $P$ is generated by a cycle of length $p$. Could you provide any ...
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1answer
127 views

Does every finite nilpotent group occur as a Frattini subgroup?

The Frattini subgroup of a finite group is the intersection of its maximal subgroups. It is well-known that the Frattini subgroup of a finite group is nilpotent. My question is whether a kind of ...
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1answer
178 views

Using the conjugacy class equation [duplicate]

Let $G$ be a group of order $p^2$. Use the class equation to prove that $G$ is abelian. The conjugacy class equation, at least how I remember it, is $$ |G| = |Z(G)|+\sum_{x\in I \backslash Z(g)} ...
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1answer
68 views

Subgroups of a finite abelian group

Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup $H$ such that $H\cong G/K$ and $H\cap K=\langle 0\rangle$? I'm not sure where to start.
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2answers
32 views

Finite groups and one-to-one functions on them.

I am having trouble with this problem: Assume that $(\mathbb{G}, *)$ is a finite group and there exists a positive integer $n$ such that gcd($n, |\mathbb{G}|)=1$. Prove that the function $F_n: ...
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1answer
72 views

Finite group $G$ is product of a subgroup $H$ and normalizer of a Sylow $p$-subgroup of $H$

Let $G$ be a finite group, $H$ a normal subgroup of $G$ and $P$ a Sylow $p$-subgroup of $H$. Let $N_G(P)$ be the normalizer of $P$ in $G$. Show that $G=N_G(P)H$.
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1answer
76 views

Sylow subgroups of soluble groups

Suppose $G \leqslant S_p$ acts transitively on $\{1,...,p\}$ for prime $p$. Let $P \leqslant G$ be a Sylow p-subgroup. Is it true that $G$ is soluble <=> $P \triangleleft G$?
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1answer
153 views

Elements of order 3 in $PGL(4,\mathbb{R})$

I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class. My thoughts: consider instead $GL(4,\mathbb{C})$ ...
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95 views

Central extensions of elementary abelian p-groups

Given an elementary abelian $p$-group $E$, we can consider $E$ as a trivial $E$-module; my first question is : How can one compute the rank of of the cohomology group $\operatorname{H}^n(E,E)$, $n ...
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2answers
60 views

primitive root of residue modulo p

I was trying to prove that for the set $\{1,2,....,p-1\}$ modulo p there are exactly $\phi(p-1)$ generators.Here p is prime.Also the operation is multiplication. My Try: So I first assumed that if ...
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0answers
60 views

Show that the number of elements in the equivalence class is n?

Question is : Let $G$ be a finiote group of order n = $p^{\alpha}$m, where $p$ is a prime number and if $p^r$ | m but $p^{r+1} \nmid$ m Let $\mathcal M$ be the set of all subset of $G$ which have ...
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0answers
50 views

About some special kinds of group automorphisms

let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$. Is there any non-trivial ...
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2answers
98 views

If $G$ is finite simple and $H \le G$ has index $2m$, then an involution of $G$ is conjugate to one in $H$

I have been trying to prove the following for a while now, with plenty of attempts and ideas, but no success. I would appreciate a gentle nudge in the right direction. Suppose that $G$ is a finite ...
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0answers
46 views

a question about fixed-point-free automorphism group 2

In this paper, Rowley (1995), there is a theorem: Let $A$ and $G$ be finite groups. Suppose that $A$ acts fixed-point-freely on $G$ and that either $A$ is cyclic or $(|G|,|A|)=1$. Then $G$ is ...
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1answer
64 views

Find the conjugacy classes of $D_6$

I am following an example in my lecture notes, but I have come to a part which I cannot get to work for myself. Thanks. Find the conjugacy classes of $D_6$. Take $$G = D_6 = \langle a,b \mid a^3 =b^2 ...
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1answer
280 views

Irreducible representations (over $\mathbb{C}$) of dihedral groups

Find number of complex irreps of the group $D_n$. Find dimension of the irreps. I know that The number of complex irreps of a finite group is equal to the number of conjugacy classes of the ...
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1answer
41 views

Property of G-modules involving the invariant elements under the G-action

I am stuck at some basic fact I would like to prove. I tried proving it using $G-$orbits and cardinalities, but without success. Let $p$ be some prime number, $G$ be a finite $p-$group and $A$ a ...
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1answer
140 views

Lower Exponent P Central Series

The lower exponent $p$-central series for a $p$-group $G$ is defined by $G=P_1(G) > P_2(G) > \ldots > P_c(G) = 1$, where $$P_i(G)=[P_{i-1}(G), G] P_{i-1}(G)^p.$$ If $G_i=G/P_i(G)$ and ...
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1answer
57 views

GF(113) arithmetic using tables?

I need to work with the Galois Field of (prime) characteristic 113. I am wondering if it is possible to implement multiplication and division using log/antilog tables (like I already do in different ...
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2answers
115 views

the nilpotency class of Frobenius kernel

As we know, if G has a fixed-point-free automorphism of order p, then G is nilpotent, can we know something about the nilpotency class of Frobenius kernel ?
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1answer
120 views

a question about fixed-point-free automorphism

Let G be a finite group with a fixed-point-free automorphism a of order 3. Prove that [x,y,y]=1 for all x,y in G.
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1answer
170 views

Finite abelian p-group and an element of maximal order

I'm studying for an exam and I'm having trouble understanding the proof given for the following statement: Suppose $G$ is a finite abelian $p$-group and $a \in G$ has maximum order, then there ...
4
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2answers
77 views

Find the center of a specific group

The group $G$ is generated by the two elements $\sigma$ and $\tau$, of order $5$ and $4$ respectively. We assume that $\tau\sigma\tau^{-1}=\sigma^2$. I have shown the following: * ...
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1answer
54 views

Composition Series of $A_4 \times S_5$

Please help me with the following question: Find the composition series of $A_4 \times S_5$ and prove that this series is indeed a composition series. Afterwards, find a group with the same ...