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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2
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1answer
36 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] ...
3
votes
1answer
47 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 ...
3
votes
0answers
23 views

A normal intermediate subgroup in L30 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 ...
3
votes
1answer
35 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, ...
0
votes
1answer
20 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 ...
1
vote
1answer
32 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 ...
4
votes
1answer
58 views

Is there a simple group and a subgroup with intermediates lattice L30?

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 $\mathcal{L}_n = ...
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0answers
61 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 ...
2
votes
1answer
75 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} ...
1
vote
1answer
31 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 ...
4
votes
1answer
46 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 ...
0
votes
1answer
21 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 ...
3
votes
2answers
242 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 $2$?
3
votes
1answer
51 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 ...
1
vote
2answers
39 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 ...
0
votes
1answer
42 views

Sylow subgroups in a group of order 112 [on hold]

Let $G$ be a group of order $112=2^47$. Prove that if a Sylow $7$-subgroup of $G$ is not a normal subgroup of $G$, then $G$ has a normal Sylow $2$-subgroup. Any comments are appreciated for me.
1
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2answers
49 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 ...
3
votes
2answers
81 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 ...
0
votes
1answer
30 views

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial [on hold]

I need example to satisfy this lemma: Let $P$ be a $p$-group and let $N$ be a nontrivial, elementary abelian normal subgroup of $P$ which has a complement $X$ in $P$. If $P = \langle y \rangle X$ for ...
4
votes
1answer
46 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 ...
3
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0answers
31 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 ...
4
votes
2answers
62 views
+50

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 $H$ ...
1
vote
1answer
27 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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vote
3answers
31 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 ...
0
votes
1answer
31 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 ...
-1
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2answers
32 views

Some equations in $\mathbb F_{37}$

How to solve efficiently the equations: 1) $[17][b]=[21]$ 2) $[17][b]=[1]$ in $\mathbb F_{37}$
5
votes
0answers
48 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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0answers
41 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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0answers
32 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 ...
2
votes
2answers
52 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
32 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?
3
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0answers
39 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 ...
2
votes
1answer
72 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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0answers
25 views

Frobenius groups of order 36 [closed]

Is there a Frobenius group of order 36? If yes, what is it's structure as semidirect product of two subgroups?
0
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2answers
62 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 ...
2
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0answers
58 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?
2
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0answers
36 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 ...
0
votes
2answers
46 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 ...
1
vote
1answer
33 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 $n_3\in\{1,4,10\}$. ...
1
vote
1answer
31 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 ...
2
votes
1answer
58 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 $B \leq A$, ...
1
vote
1answer
12 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 ...
4
votes
2answers
85 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
votes
1answer
112 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
votes
1answer
105 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 ...
3
votes
2answers
38 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 ...
2
votes
1answer
37 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 ...
2
votes
1answer
43 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), ...
2
votes
0answers
61 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.
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3answers
35 views

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd , a homomorphism?

Is the map $f:S_n \to A_{n+2}$ given by $f(s)=s$ , when $s$ is even and $f(s)=s \circ (n+1 \space , \space n+2)$ , when $s$ is odd an injective homomorphism ? I can show that if it is a homomorphism ...