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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Automorphisms of finite almost simple groups

Let $P$ be a finite nonabelian simple group. Let $G$ satisfy $$ P\leqslant G \leqslant {\rm Aut}(P), $$ where $P$ is identified with $\rm{Inn}(P)$. I am trying to see if $$ {\rm Aut}(G)\cong ...
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3answers
87 views

How does one enumerate $\mathrm{Aut}(G)$, or at least compute $|\mathrm{Aut}(G)|$?

I know that the automorphisms in $\mathrm{Aut}(G)$ preserve the order of elements of $G$, so if $G$ is partitioned according to $\mathrm{Ord}$ (order), the product of the cardinalities of the ...
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2answers
31 views

In the group A/B find the order of the coset (x$_1$+2x$_3$)+B

A is an abelian free group, with the base x$_1$,x$_2$,x$_3$. will be B a sub-group that created with x$_1$+x$_2$+4x$_3$,2x$_1$-x$_2$+2x$_3$. In the group A/B find the order of the coset ...
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1answer
100 views

Group exponent properties

Is it true that for any finite cyclic group $G$, it holds that $\operatorname{exp}(G)=|G|$? My first thought was yes, since if $G$ is cyclic, then we know it has an element of order |G|, and since ...
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2answers
42 views

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.
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2answers
76 views

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even.

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even. I fill that i have a way to prove it: the sign of $\sigma$ is $-1$. so $(-1)^{n-t}=-1$, when $t$ is the ...
4
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1answer
97 views

If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian.

Let $G$ be a finite group which possesses an automorphism $σ$ such that $σ(g)=g$ if and only if $g=1$. If $σ^2$ is the identity map from $G$ to $G$, prove that $G$ is abelian. This is what I got ...
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3answers
45 views

How many subgoups are there in $\mathbb Z_{24}$? [closed]

How many subgoups are there in $\mathbb Z_{24}$?
0
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3answers
113 views

Show that $G\{m\}$ is finite for all non-zero $m\in \mathbb{Z}$

Let $G$ be an abelian group, and let $m$ be an integer, then we define $G\{m\} := \{a\in G:ma=0_G\}$. Now, suppose that $G$ is an abelian group that satisfies the following properties: (i) For all ...
4
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2answers
65 views

Existence of nontrivial unit in $\mathbb{Q}[G]$, where $G$ is finite.

Suppose $G$ is a finite group of order $|G|>1$, and $\mathbb{Q}[G]$ is the group ring. I'm curious about an example of a nontrivial invertible element, i.e., one that is not of the form $ag$, ...
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2answers
63 views

Prove that a subset $\{a \in A \mid \chi (a) = 1\}$ is an $(n - 1)$-dimensional subspace of $V$.

Let $A$ be the additive group of n-dimensional vector space $V$ over the field $\mathbb{F}_p$. Let $\chi$ - a nontrivial irreducible complex character of A. Prove that a subset $\{a \in A \mid ...
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2answers
30 views

Question about number of elements in $S_p$ and number of $p$-sylow groups.

Let $G=S_p$ where $p$ is a prime. How many elements with order $p$ in $G$, and what are they? How many $p$-sylow their is in $G$? I will be glad to see a simple solution.
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5answers
102 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
61 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). ...
0
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2answers
36 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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0answers
37 views

Definition of K-normal subgroups

"We say that a group K acts semisimply on an abelian group A, if the intersection of all maximal K-normal subgroups of A is trivial" from "Construction of Finite Groups" article by "BESCHE and EICK" ...
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2answers
77 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 ...
3
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1answer
34 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
33 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
57 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
36 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
92 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
91 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
37 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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votes
2answers
64 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
41 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
46 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
68 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
41 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
32 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. ...
4
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1answer
56 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
45 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
68 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 ...
3
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0answers
61 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
99 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
130 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
22 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
46 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
51 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
83 views

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

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
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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
71 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
56 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
62 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
27 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
78 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
38 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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0answers
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Visual Solution - Find All (Cyclic) Subgroups of $D_4$ generated by 1, 2, … elements - Fraleigh p. 84 8.19

Verify that the subgroup diagram for $D_4$ shown in Fig. 8.13 is correct by finding all (cyclic) subgroups generated by one element, then all subgroups generated by two elements, etc. Here, $p_i$ mean ...
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0answers
29 views

Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
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1answer
95 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, ...