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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Group theory and centres [duplicate]

If $G$ is a $p$-group and $H$ is a non trivial normal subgroup of $G$, how do I show that the size of $H\cap Z(G)$ (where $Z(G)$ is the centre of $G$) is $\ge p$? A hint is given to consider the ...
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57 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...
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1answer
42 views

what is the maximom order of an element is $\mathbb S_{15}$ [duplicate]

Let $S_n$ be the permutation group of order $n$. What is the maximom order of an element on $\mathbb S_{15}$. Is there any way to find maximom order of an element in $\mathbb S_{15}$ or find the ...
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21 views

A question about finite simple groups of Lie type

I have a question about finite simple groups of Lie type. By the classification theorem of finite simple groups, we know that there are four important class of finite simple groups as follows: ...
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1answer
116 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
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2answers
129 views

Order of the Rubik's cube group

Associated to the Rubik's cube is a group as described in this Wikipedia article: $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise ...
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1answer
73 views

A normal intermediate subgroup in $B_3$ lattice?

Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups between $H$ and $G$. An intermediate subgroup $H \subset K \subset G$ is a ...
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1answer
43 views

When a normal subgroup $N$ admits a complament?

Let $G$ be a finite group and let $N$ be a normal subgroup. I am looking for conditions on $N$ (and maybe also on $G$) such that there exist a subgroup $H$ of $G$ such that $$G=N\rtimes H.$$ Clearly, ...
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1answer
40 views

Number of subgroups and normal subgroups

I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example: How many normal subgroups does S3 have? How many subgroups of order 4 has group S4? And does ...
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1answer
37 views

What is the forgetful functor for this simple finite category (based on the cyclic group C3)?

I am trying to understand what a forgetful functor is. I have read the definition here but I still cannot figure out what the forgetful functor is exactly for the simple cyclic group C3 viewed as a ...
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1answer
94 views

Is there a simple group and a subgroup with intermediates lattice $B_{3}$?

Let $G$ be a finite simple group and $H$ a subgroup. We consider the lattice of intermediate subgroups between $H$ and $G$, noted $\mathcal{L}(H \subset G )$. Let $B_{n} = \mathcal{L}(\{ e\} ...
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67 views

Finite groups and manifolds

my question is: can we connect finite groups with algebraic manifolds as: Take for example Togliatti surface $X$ with set of 31 singular points $P$. Then consider action $Aut(X)$ on $P$, then group ...
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1answer
97 views

Presentation of a group isomorphic to $A_4$

I have a group $G$ defined by $G = \langle x,y,z|x^2 = y^3 = z^3 = xyz \rangle$ and we know that $a$ $=$ $xyz$ belongs to the centre of $G$. But im struggling to show that $\frac{G}{\langle a\rangle} ...
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1answer
34 views

How many possible isomorphisms do we have between G and H? [duplicate]

Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order ...
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1answer
60 views

Sylow subgroups of Symmetric Group

The symmetric group (=permutation group) $S_n$ acts on the set $X_n$ of polynomials in $n$ variables $x_1, x_2, \cdots, x_n$ [with coefficients from $\mathbb{Z}/ \mathbb{Q}/$ or any ring of ...
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1answer
27 views

Groups of order 36 - another step in lemma 5.4.

This is a follow up to my question last night Groups of order 36 where I was confused about the first step of Lemma 5.4 of http://matwbn.icm.edu.pl/ksiazki/fm/fm92/fm9211.pdf. I am now confused about ...
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2answers
362 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
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1answer
70 views

Group of order 396 isn't simple

Prove that group of order $396=11\cdot2^2\cdot3^2$ is not simple. $n_{11}$ is $1$ or $12$, so I assumed $n_{11}=12$ and tried to look at the action of the group on $Syl_{11}\left(G\right)$ by ...
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2answers
43 views

Relationship between submonoids and subgroups

I'm taking my first abstract Algebra course. During one of the last lessons, my teacher told us that If $G$ is a group and $M$ is a finite submonoid of $G$, then $M$ is a subgroup of $G$. For the ...
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2answers
59 views

Groups of order 36

Prove: If a group $G$, of order 36 has a subgroup of order 18 ,$H$, then $G$ either has a normal subgroup of order 9, or a normal subgroup of order 4. This came about while reading the same article ...
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90 views

Order of any element divides the largest order.

Let $A$ be a finite Abelian group and let $k$ be the largest order of elements in A. Prove that the order of every element divides $k$. This is my attempt, I sense there is something wrong\incorrect ...
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1answer
67 views

Group of order $pqr$ and cyclic subgroup

Let $G$ be group of order $pqr$, when $p,q,r$ are different prime numbers. Does $G$ must have normal cyclic subgroup $H$ such that $G/H$ is cyclic too ? I know that $G$ has normal sylow subgroup of ...
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45 views

How many squares in a finite group?

Let G be a finite group and denote by S[G] the number of squares in G. The maximum, S[G]=n, is attained for a group of odd order n since each element has a square root in that case. At the other ...
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3answers
99 views

Let $H$ be a subgroup of the group $(R, +)$ such that $H$ $∩$ [-1,1] is a finite set containing a non zero element. Show that $H$ is cyclic.

Observations: Since $H$ is a subgroup of $(R, +)$ so $0 \in H.$ If $1 \in H,$ then all positive integers belong to $H.$ But $H$ is closed wrt addition, so the negative integers must belong to $H$ ...
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1answer
36 views

permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$,

Let $H$ be a subgroup of $G$ and $N$ a normal subgroup of $G$. permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. ...
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3answers
38 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
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1answer
44 views

class equation of order $10$

Is it a class equation of order $10$ $10=1+1+1+2+5$. As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is ...
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2answers
38 views

Some equations in $\mathbb F_{37}$

How to solve efficiently the equations: 1) $[17][b]=[21]$ 2) $[17][b]=[1]$ in $\mathbb F_{37}$
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64 views

Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has ...
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43 views

Direct products of simple non-abelian groups 2

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ a simple non-abelian subgroup of $G$ such that $Z(G)\leq N$ and $\frac{G}{N}\cong A$ be a non-abelian simple group. Is it true that $G=N\times A$ ...
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38 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
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2answers
55 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
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2answers
39 views

cyclic groups -class is prime number

How can I prove that a group $G$, such that $|G| = p$, where $p \in \mathbb{P}$, is cyclic?
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55 views

Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$

I am working on this problem with lots of nesting definitions: Show that $G/Z(R(G))$ is isomorphic to a subgroup of $Aut(R(G))$. For your info, $R(G)$ is called the Radical of $G$, defined as ...
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1answer
111 views

Online Archive of Master Thesis

I am thinking about taking the thesis route to complete my master in pure math. In anticipation of these in the coming semesters, here are my questions: (1) Do you know of any links to archive of ...
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76 views

How do I generate group table for elliptic curves over finite fields

Can someone please explain how to generate a group table for an elliptic curve over a finite field? The number of solutions or points are about 16 and it is not possible to do them by adding each ...
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0answers
68 views

a finite group of order $360$ never has an irreducible character of degree $15$

The following problem is an exercise in character theory: "Prove that a finite group of order $360$ never has an irreducible character of degree $15$." Could you help me for it?
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41 views

Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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2answers
49 views

What is the difference between $[H, g]$ and $[h, g]$?

I am working on this problem, where $[H, g]$ is the commutator group: Let $H$ be a subgroup of $G$, show that $[H, g] = [H, \langle g \rangle]$. Before solving it, I need to understand the ...
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1answer
60 views

calculate the number of sylow p subgroups of a5

Calculate the number of Sylow $p$-subgroups of $A_5$ We have $|G|=60=2^2\cdot 3\cdot 5$ Let $n_p$ be the number of Sylow $p$-subgroups of $G$. By Sylow's third theorem, we have $n_3\in\{1,4,10\}$. ...
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1answer
34 views

In general, what ways are there to show if 2 groups are isomorphic?

I take it that if the number of elements of a given order n is not the same between 2 groups, then they are definitely not isomorphic. So for example if I need to show that $C_{25}$ is not isomorphic ...
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1answer
63 views

Proof Using Lagrange's Theorem

I am working on a problem in Kurzweil & Stellmacher's introductory finite group theory that looks like this: Let $A, B$, and $C$ be subgroups of the finite group $G$. Prove that if $B \leq A$, ...
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1answer
16 views

A finite group of isometries is isomorphic to a subgroup of $SO(3)$?

I want to show that the rotation group of a polyhedron is isomorphic to a finite subgroup of $SO(3)$, since then I can use the classification of those subgroups to classify all polyhedral rotation ...
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2answers
96 views

Finite cyclic subgroups of $GL_{2} (\mathbb{Z})$ [duplicate]

How could we prove that any element of $GL_2(\mathbb{Z})$ of finite order has order 1, 2, 3, 4, or 6? I am aware of the proof supplied here at this link: ...
3
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1answer
137 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about ...
3
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1answer
112 views

Factorizing elements of a group into a product of generators.

$$ s = (1\ 2) \\ t = (1\ 2\ 3\ ...\ n) $$ Given the Symmetric Group $S_n$ generated by $s$ and $t$, is there a way to quickly factor an element $g \in S_n$ into a minimal product of positive powers ...
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2answers
39 views

Prove that $S= \langle(1,2,3,4) \rangle$ has 3 conjugates in $S_4$

Some things I know: $S = \{ (1),(1,3)(2,4), (1,2,3,4),(1,4,3,2)\}$ $(2,4) \in N_G(S)$ Number of conjugates = $[G: N_G(S)]$ This seems like such a easy question but it made me realised that I do ...
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1answer
41 views

Question on simple subgroup $H$ and a normal subgroup $N$, of $G$

This one is a bit strange to me, mainly the third hypothesis. It goes as follows: Given a group (finite) group $G$, and $N, H \leq G$ such that $N$ is normal in $G$, and $H$ is simple ...
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1answer
46 views

Finite group with unique subgroup of each order.

Let $G$ finite group, and suppose $G$ has unique subgroup of each order (which divides $G$'s order) - Show that $G$ is cyclic. I reduced the problem to sylow subgroups of $G$ (they are all normal), ...
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1answer
75 views

How unitarize an irreducible representation of a finite group?

Let $G$ be a finite group acting irreducibly on the space $V$. $$\psi : G \to Aut(V)$$ Question: How unitarize the representation $V$? I'm looking for a computable process.