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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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 ...
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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 ...
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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 ...
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3answers
244 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 ...
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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 $$ (ab)^3 = a^3 b^3 $$ ...
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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 ...
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124 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 ...
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70 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 ...
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1answer
150 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 ...
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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 ...
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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].
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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 / ...
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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?
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0answers
326 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 ...
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2answers
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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 ...
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1answer
38 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 ...
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120 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 ...
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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 ...
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1answer
89 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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106 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
184 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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180 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
314 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
58 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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162 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
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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
79 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
136 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
142 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
73 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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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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31 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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82 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 ...