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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5answers
189 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
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2answers
77 views

Permutation Group-$S_{10}$

How many elements of order $30$ are there in the symmetric group $S_{10}$? I worked out and got $10500$ - using Computations and adjusting each individual cycle's position ( $2-3-5$ , $3-2-5$ etc). ...
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2answers
51 views

Isomorphism class of $\mathbf{U}(p^n)$

Note that $\mathbf{U}(k)$ is the unitary group. i.e. $\mathbf{U}(k)=\{x<k | \gcd(x,k)=1\}$ We need to find the isomorphism class of $\mathbf{U}(p^n)$ where $p$ is an odd prime. The isomorphism ...
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3answers
668 views

Product of disjoint cycle

I've found the question to find product cycle of let be $\phi$ = (2, 4, 9, 7,) (6, 4, 2, 5, 9) (1, 6) (3, 8, 6) in S9. I know how to express the permutation of S9 as product of disjoint cycle, like ...
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1answer
46 views

How to find the number of transposition

I just learning the abstract algebra now, I'm stuck to find how many transpositions can be made from $(1\ 8)(2)(3\ 6\ 4)(5\ 7)$?
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1answer
41 views

How to find how many cosets are of $H \cap K$?

I'm confuse to find how many cosets of $H \cap K$ are in the G? If $G$ is a group of order 48, then $H$ of order 8, $K$ of order 6, <= $G$.
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1answer
155 views

Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
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0answers
38 views

It is true that $\mathrm {Im}(f^{n_{0}})=\mathrm {Im}(f^m)$ for all $m\geq n_0$ implies $\mathrm {Im}(f^{n_{0}})=\{0\}$

Let $G$ be a finite abelian group, and $f: G\longrightarrow G$ an endomorphisme of $G,$ such that $\ker(f)\neq \{0\},$ and $\mathrm {Im}(f)$ a propre subgroup of $G,$ so we have a descending chain ...
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2answers
268 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
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3answers
123 views

How many Homomorphisms can be between $Z_6$ to $Z_{18}$?

How many Homomorphisms can be between $Z_6$ to $Z_{18}$? and the most important: What is the algorithm for calculating, step by step?
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1answer
40 views

Subgroups of $\mathbb{D}_6^n$

Let $\mathbb{D}_6=\{1,x,x^2,y,xy,x^2y\}$ be the Dihedral group of order 6. I'm trying to find two subgroups $N\le \{1,x,x^2\}^n$ and $M\le \{1,y\}^n$ such that $MN=NM$ (so that the product is also a ...
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2answers
89 views

Fast way of calculating the order of an element in $\mathbb{Z}_n$?

Is there a fast way of calculating the order of an element in $\mathbb{Z}_n$? If i'm asked to calculate the order of $12 \in \mathbb{Z}_{22}$ I just sit there adding $12$ to itself and seeing if the ...
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1answer
66 views

Unique Complete Reducibility of Finite Groups

Maschke's Theorem states that every complex representation $(\rho,V)$ of a finite group $G$ can be written as a direct sum of irreducible representations that form subsets of V, such that $V = ...
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1answer
100 views

Consequences of Schur's Lemma

Schur's Lemma states that given two irreducible representations $(\rho,V)$ and $(\pi,W)$ of a finite group $G$ (V and W are linear vector spaces over the same field), and a homomorphism $\phi\colon ...
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1answer
94 views

Stitching of Coset Diagrams

Can any one assist me to give me concept of Handles in the coset diagram? How do we identify it and how can we make new presentaions by joining the handles of the coset diagrams of distinct groups? ...
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2answers
50 views

What does it mean to divide groups?

Here a group is defined as division (?) of groups: $$G=GF(q^{n+2})^*/GF(q)^*,$$ where $GF(q)^*$ is the multiplicative group of Galois field's GF(q) non-zero elements. What would the $G$ contain? ...
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1answer
56 views

If $\sigma\in S_n$ is of length $n$, then $\sigma$ generates its centralizer

Let $S_n$ and $C_G(\sigma)$ denote the symmetric goup and the centralizer of $\sigma\in S_n$, respectively. I want to show: If $\sigma$ is of length $n$, then $C_G(\sigma)=\langle\sigma\rangle$, i.e. ...
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1answer
152 views

Reading a Character Table in Magma

These are the outputs of two character tables from Magma. The first is for $A_5$. The second is for $GL_3(2)$. What is the significance of the '$+$' and '$0$' symbols? I can produce more tables if ...
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1answer
79 views

If two powers of permutations are equal and have no common symbols, they're the identity. - Mulholland p. 44 Proof to Theorem 4.2

Theorem 4.2 (Order of a Permutation): The order of a permutation written in disjoint cycle form is the least common multiple of the lengths of the cycles. Proof: One cycle: As we noted above, a ...
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1answer
180 views

representations of the dihedral group

Let $\rho_\epsilon(a)=\begin{bmatrix}\epsilon & 0\\0 & \epsilon^{-1}\end{bmatrix}$ and $\rho_\epsilon(b)=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$ I can prove that $\rho_\epsilon$ is ...
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0answers
84 views

Fusion of elements of order 8 in a finite simple group

Must a finite group $G$ with an element $x$ of order 8 such that $x^2$ is conjugate to $x^{-2}$ but $x$ is not conjugate to any element of $\{x^3,x^5,x^7\}$ also have a normal subgroup of index 2? I ...
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2answers
339 views

Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
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2answers
205 views

Exercise 5C10 in Isaacs' Finite Group Theory

Problem: Suppose that $G$ is simple group and has an abelian Sylow $2-$subgroup of order $8$. Show that the order of $G$ is divisible by $7$. Is there any hint to solve this problem? I'll be glad if ...
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1answer
29 views

irreps of $p^3$-group is faithful representation

Let $A$ be an irreps of $p^3$-group. Prove that $A$ is faithful representation. I know that $p^2$-group and $p$-group are abelian. I have to show, that $Ker A=e$ I have no idea how to start it
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1answer
72 views

How does Cauchy's theorem follow from Sylow's theorem?

Very quickly, Sylow's first theorem says a sylow p-subgroup of order $p^rm$ exists and Cauchy's theorem says if $p \vert |G|$ then there is an element of order $p$. It's often said that Cauchys ...
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1answer
166 views

Finite groups with unique minimal subgroup

Let $G$ be a finite group. Let $G$ has a unique non trivial minimal subgroup. Then $G$ is a p-group. How to prove the theorem which says that: If $G$ has a unique non trivial minimal subgroup and if ...
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4answers
237 views

Order of conjugate of an element given the order of its conjugate

Let $G$ is a group and $a, b \in G$. If $a$ has order $6$, then the order of $bab^{-1}$ is... How to find this answer? Sorry for my bad question, but I need this for my study.
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1answer
109 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
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4answers
85 views

How many isomorphism of $\phi :\mathbb Z_{4} \rightarrow \mathbb Z_{4}$?

I'm interested in how to find it, not the answer itself. I'm confuse to solve this question, I know isomorphism is bijective, and in this case it called Automorphism. But, I can't find a way how to ...
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1answer
81 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
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2answers
73 views

if $G$ is a group of order $p^n$ where $p$ is prime

If $G$ is a group of order $p^n$, where $p$ is prime and $n \geq 1$, prove that $G$ must have a subgroup of order $p$.
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1answer
30 views

$\phi : \mathbb{Z}_5 \to $ H is a homomorphism, where H is a 5 order group .

If $\phi(1) = a^3$, then $\phi(4)$ is ...? How to get the answer correctly, I'm still beginner in abstract algebra.
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1answer
245 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
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1answer
52 views

decompose a real representation of a group $C_2\times C_2

Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $ \Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and $ ...
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1answer
131 views

Is there an infinite group that contains every finite group (and no infinite group) as a subgroup?

Question is in the title. For bonus points, construct the group $G$ such that it also has no infinite proper subgroups. (This second question relates to the Prüfer group, but that group is abelian, ...
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3answers
782 views

Given a group, how to show the distributive law and some examples (does the distributive law have to be an axiom?)

tl;dr: What properties does a set need for it to have the distributive law? Does it need to be an axiom? Given a group (i.e. satisfies the closure, associative, inverse and identity ...
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1answer
105 views

subgroups of finite cyclic group

Let $G=(g)$ be a finite cyclic group generated by $g$ with $|G|=n$, and let $d \in \mathbb{N}$ with $d|n$, then an unique subgroup $H$ of $G$ with $|H|=d$ exists. Proof of existence: $\exists m \in ...
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1answer
136 views

Connections and Differences between these Cayley Diagrams for $A_4$ and $S_4$ - Carter pp. 80, 82

Reference: Nathan Carter pp. 80, 82, ch. 5, Visual Group Theory Figure 5.24. As you will read in the next section, it is no coincidence that [the Cayley digram for $S_4$] looks cube-like. A Cayley ...
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1answer
50 views

$S_n$ acting on $\{1\;…\;n\}\times \{1\;…\;n\}$

Let $X=\{1,\;...\;n\}$ and $S_n$ act transitively on $X\times X$ i.e. $s:\;(m,n)\mapsto (s(m),s(n))$. Compute the orbits under this action. Attempt: I claim that there are only two orbits, ...
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2answers
63 views

On maximal subgroup's index of a finite group

Let $G$ be a finite group and $p$ be a prime. Assume that for every maximal subgroup $M$ of $G$ we have $[G:M]$ is not $\equiv1$ mod $p$. Prove that $G$ has a normal Sylow $p$-subgroup. ...
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0answers
231 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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2answers
63 views

If the product $x_1x_2…x_n=1$, prove that each $x_i=1$.

Let $G$ be a finite solvable group and let $x_1,x_2,...,x_n$ be elements of $G$ of pairwise relatively prime orders. If the product $x_1x_2...x_n=1$, prove that each $x_i=1$. I have no idea. Tell me ...
3
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1answer
73 views

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$.

Let $G$ be a finite group and let $H$ be a solvable subgroup of $G$. Let $N=N_G(H)$ and assume that $N/H$ is a nonabelian simple group. Prove that $N=N_G(N)$. $N=N_G(N)$. That means ...
3
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1answer
111 views

Number of subgroups of order 48 in $\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$

I thought it would have sufficed to show that every subgroup of G=$\mathbb{Z}_{8} \oplus\mathbb{Z}_{12}$ must be formed by couples (a,b) whose set of a's and the set b's form a subgroup of ...
2
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1answer
92 views

Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$.

Let $G$ be a finite group and $p$ be a prime. Let $H$ be a subgroup of $G$ which contains $N_G(P)$ for some Sylow $p$-subgroup $P$ of $G$. Suppose $P \subseteq H^g$ for some $g \in G$. Prove ...
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0answers
44 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
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1answer
116 views

A question about perfect group

Let $G$ be a finite group. Show that if $G = G'$, then $Z$$\left( G/{Z\left( G \right)} \right)=1$. My attemp is here. Fact: $G' \le N$ if the quotient group $G/N$ is abelian. Since $G = G'$, by the ...
5
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1answer
152 views

Prove that if $A \vartriangleleft G$ is abelian, then $A$ has a complement in $G$.

Let $G$ be a finite group. Suppose that the intersection of all of the maximal subgroups of $G$ is trivial. Prove that if $A \vartriangleleft G$ is abelian, then there exist $U \subseteq G$ such that ...
3
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1answer
260 views

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. [closed]

Let $G$ be a finite group. Suppose $N,H,K \subset G$ are subgroups such that $NH=G$ and $(N \cap H)K=G$. Prove that $N(K \cap H)=G$. I have no idea. Give me some hints.
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2answers
183 views

An exercise about simple groups

Let $G$ be a finite group, $D=\{(g,g):g\in G\}$ is a subgroup of the direct product $G\times G$. Show that $G$ is simple if and only if $D$ is a maximal subgroup of $G\times G$. I tried to prove by ...