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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4
votes
2answers
143 views

Show that $G$ ( subgroup of $\mathrm{GL}(E)$) is finite.

I came across with, I think, a difficult problem : Let E a Hermitian space with a Hermitian norm $||\ ||$. We provide $\mathcal{L}(E)$ with the norm $|||\ \ |||$ subordinated to $||\ ||$. ...
1
vote
0answers
33 views

$ K \rtimes_{\phi_1} C \cong K \rtimes_{\phi_2} C$ when $\phi_1(C), \phi_2(C)$ are conjugated and $C$ is a product of two cyclic groups

I know the following result: Let $C$ be a finite cyclic group, $K$ a finite group such that there exist homomorphisms $\phi_1,\phi_2$ $\phi_i:C \to Aut(K) $ such that $\phi_1(C), \phi_2(C)$ are ...
0
votes
1answer
42 views

Is this object a group?

$\begin{array}{ccccccccc} \times&e_0&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ e_0&I&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ ...
5
votes
1answer
82 views

Does $\mathrm{GL}_{n-2}(\mathbb{Z})$ has an element of order $m$?

Let me introduce the context: A few week ago I have made the following contest as a "homework" : ENS contest (France) 2006 which is essentially about $SL_n(\mathbb{Z})$ group, finite subgroup of ...
0
votes
1answer
59 views

Summing the traces of matrix powers

Let $G=\langle h\rangle_n\subset{\rm GL}(m,\mathbb{C})$ be a cyclic group of order $n$. I wonder if there is a good formula for calculating the sum $\sum_{g\in G}{\rm Tr}(g)$ via ${\rm Tr}(h)$, for ...
1
vote
1answer
23 views

Can't remember definition of $\lvert G \rvert_{p'}$

For $G$ a finite group, I know that $\lvert G\rvert$ denotes the order of the group. My question is: What is $\lvert G\rvert_{p'}$? Also is this the same as $\lvert G\rvert_p$ (without the prime on ...
0
votes
1answer
28 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
1
vote
0answers
36 views

Let $G$ be a group of order 24 and suppose $n_2(G) > 1 \ \ and \ \ n_3(G) > 1$ . Then $G \cong S_4$

My attempt is : Since $n_3 > 1$ and $n_3 \equiv 1 \ \ mod \ \ 3 $ and divides 8, then the only possibilty is $n_3 = 4$ and thus $| G:N| = 4$, where $N = N_G(P)$ and $P \in Syl_3(G)$. Then $G/K $ ...
5
votes
2answers
121 views

Why free presentations?

What is the motivation to study "free" presentations of groups,even though all (or almost all) the questions (or the problems) concerning this type of presentations are known to be undecidable ?
1
vote
2answers
233 views

$g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
0
votes
1answer
141 views

Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
3
votes
1answer
42 views

Having trouble grasping the class equation as an explanation as to why a conjugate class's order divides the order of a group.

Suppose $|G|$ is a prime power $p^n$ and that $N$ is a normal subgroup of $G$. Show that $|y^G|$ is a power of $p$ whenever $y \in G$ Attempt: Firstly, I assume that $y^G = \{ gyg^{-1} | g \in G ...
1
vote
1answer
58 views

Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook ...
3
votes
1answer
57 views

Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. ...
1
vote
1answer
43 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
0
votes
1answer
27 views

Show that the orbit of $H$ containing $x$ is equal to the right coset $Hx$

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $X$ be the set of elements of $G$. Let $ \ast : H \times X \to X$ be given by $$ h \ast x = hx (h \in H, x \in X)$$. QUESTION: Let $x \in X$. ...
2
votes
2answers
99 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
1
vote
1answer
35 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
1
vote
2answers
53 views

Product of cyclic groups

How can you quickly tell that the product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_3$ has a 2-subgroup containing an element of order 4? Also, I don't understand the notion of multiplying ...
0
votes
1answer
54 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
1
vote
1answer
55 views

What about the index of this subgroup? [duplicate]

Let $G$ be a group, and let $H$ be a subgroup of finite index in $G$, and let $N \colon = \cap_{x \in G} \ xHx^{-1}$. Then $N$ is clearly a subgroup of $G$ which is contained in $H$ and such that ...
4
votes
3answers
251 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
3
votes
0answers
48 views

How to solve this problem on finite groups? [duplicate]

Let $G$ be a finite group whose order is not divisible by $3$ and such that $(ab)^3 = a^3 b^3$ for all $a$, $b$ in $G$. Then can we determine if $G$ is abelian or not? Since $$ (ab)^3 = a^3 b^3 $$ ...
0
votes
1answer
35 views

Determinantal order of character of a group.

The notion of determinantal order can be found in 'Character Theory of finite groups' by I Martin Isaacs. If $\chi$ be a linear character of a finite group G, show that the order of $\chi$ in the ...
4
votes
1answer
125 views

Order and element set of the group with presentation $\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$

If we have a group presentation $G=\langle a,b \;|\; a^9=1, b^3=a^3, [a,b]=a^3\rangle$, how we will get the following values: The order of the group. The elements of the group written in terms of ...
1
vote
1answer
71 views

Building a partial injective relation

Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$. Example: $A ...
2
votes
1answer
155 views

Can someone please explain the word problem (from group theory) in Calculus III layman's terms

I'd appreciate it if someone could offer a schematic explanation of the word problem from group theory in terms which someone at a calculus III level (but without any background in group theory or ...
2
votes
1answer
27 views

Permutations, cycles and conjugacy

Let $u \in S_n$ be a cycle, where $S_n$ is the group of permutations of the set with $n$ elements. Let $\sigma \in S_n$ such that the support of $\sigma \circ u \circ \sigma^{-1}$ is the same as the ...
2
votes
0answers
29 views

Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
9
votes
6answers
905 views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
6
votes
1answer
125 views

Is $\langle a,b\; |\;a^7 = 1, ab = b^3a^3\rangle$ finite?

I've been playing a little with group definitions to see what kind of things I can make up. I'm struggling to prove that this group is finite. Can anyone point me in the right direction?
2
votes
0answers
336 views

Find order of given factor group

I'm trying to find the order of this factor group: $$(\mathbb Z_{12}\times\mathbb Z_{18}) / \langle (4,3)\rangle.$$ The order of the factor group is just the number of elements in it (aka the ...
0
votes
2answers
58 views

Computing the index $[G:H]$ with $H \triangleleft G$.

As an excercise I am investigating the symmetric group $S_n$ beginning from its conjugacy classes and then taking their union to form normal subgroups. Since conjugacy classes $C_m$ contain elements ...
1
vote
1answer
40 views

Extensions of Abelian groups to non-Abelian groups

Let $N$ and $M$ be two finite Abelian groups. Is there a nice way to characterize all extensions of $M$ by $N$? I have seen a few sources where Abelian extensions of Abelian groups are discussed but ...
1
vote
2answers
125 views

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism on $g$, $g(N) =N$.

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism $g$, $g(N) =N$. I can prove this for the case when there is a subgroup $H$ with the same ...
1
vote
2answers
61 views

proving to see that a normal subgroup is equal to a subgroup if one of the subgroup is the identity.

Can anyone check my attempt on the question which i have prosed hours ago . Question: Let $G$ be a finite group and $H◁G$ a normal subgroup. Prove that $|G/H|=|G|$ if and only if $H=\{e\}$. My ...
4
votes
1answer
91 views

Groups of order $p^aq^br^c$ containing elements of order $pq$, $qr$, and $pr$, but not $pqr$

Let $G$ be a solvable group of order $p^aq^br^c$ (for distinct prime $p,q,r$) containing elements of orders $pq$, $qr$, and $pr$, but no element of order $pqr$. Furthermore, assume that $G$ is ...
3
votes
1answer
44 views

Does G necessarily have a subgroup H…

I'm confused on an abstract math question. Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup H such that $H≅G/K$ and $H∩K=⟨0⟩$. I think it is ...
0
votes
2answers
58 views

How to show that a group is finite and also normal

Let $G$ be an finite group and $H$ normal subgroup of $G$. Show $\left|G\big/H\right|=\left|G\right|$ if and only if $H=\{e\}$. Firstly I do not know how to show that $G$ is finite. Next I know that ...
1
vote
1answer
93 views

elementary row operations

We know that the elementary row operations generate the general linear group. Suppose that we have a subset of elements of a given general linear group. Is it possible to generate given general linear ...
20
votes
2answers
557 views

Can we uniquely determine a group given the orders of its elements?

Given a finite group $G$ and its order, consider a scenario in which we also know the orders of each of its elements. Does this information alone uniquely determine the group? If not, can we at ...
1
vote
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. ...
1
vote
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 ...
2
votes
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?
1
vote
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 ...
3
votes
3answers
188 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 $x\neq e$?
1
vote
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 ...
4
votes
2answers
183 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 ...
3
votes
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 ...
3
votes
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 & ...