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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7
votes
1answer
131 views

$D_8$ as a derived subgroup

Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them ...
2
votes
0answers
166 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
0
votes
1answer
70 views

Abstract orbits stabilizers

Consider $D_8$ acting on itself by conjugation. Find orbits and stabilizers for all elements of $D_8$. $$D_8=\{1,r,r^2,r^3,b,br,br^2,br^3\}$$ So far I have the orbits: {$1$}, {$r^2$}, {$r,r^3$}, ...
0
votes
2answers
234 views

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?

How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
3
votes
1answer
137 views

Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
3
votes
1answer
128 views

Dihedral groups and normal subgroups

Consider $D_8$ ={$1,r, r^2, r^3,b,br,br^2,br^3$} and the subgroup, $H$={$1,r^2$} and $K$={$b$} of $D_8$ I really need some help with these particular problems. Show that $H\lhd$$D_8$, but ...
1
vote
4answers
78 views

The isomorphism from $S_3/\langle (123)\rangle$ to $\mathbb{Z}_2$.

Suppose that $N=\{(123), (132), \operatorname{e}\}$ and $N$ is normal in $S_3$. Show that the quotient group $S_3/N$ is isomorphic to $\mathbb{Z}_2$. What mapping should I use?
1
vote
2answers
154 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
4
votes
3answers
461 views

Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$, where $G$ is abelian and $\mathbb{T}$ is the circle group.

Suppose $G$ is an abelian group and $a\in G$ and $$f:\left<a \right>\to\Bbb T$$ is a homomorphism. Can $f$ be extended to a homomorphism on $G$: $$g:G\to \Bbb T$$ ? $\Bbb T$ is the circle ...
0
votes
4answers
184 views

Can a group of order 3000 be a simple group?

Can a group of order 3000 be a simple group? How about the case of a group of order 1000?
1
vote
2answers
131 views

Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements

So I supposed $|H \cap K|>1$ $\Rightarrow |HK||H \cap K|> |HK|$ Which eventually implied that $\Rightarrow |H \cap K|>|G|$ Thus since G is a group, and H and K are subgroups then the ...
2
votes
1answer
697 views

Let G be a finite group and let H and K be subgroups of G. Suppose [G:K] and [G:H] are relatively prime. Prove G=HK [duplicate]

So I am rather confused on where to start this proof so all I've got is $[G:H \cap K]=[G:H][H:H \cap K]$ $[G:H \cap k]=[G:K][K:H \cap K]$ Thus that implies $[G:H][H:H \cap K]=[G:K][K:H \cap K]$ ...
3
votes
1answer
60 views

order of $\langle (123) , (234) \rangle$

As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ . I've started doing all the multiplications between the elements, and I've ...
2
votes
1answer
428 views

Computing $\bmod$s with large exponents by paper and pencil using Fermat's Little Theorem.

I'm having a bit of trouble computing $\bmod{mod}$s of large numbers using Fermat's Little Theorem. For example, how would you compute $7^{435627650}\mod 13$? The solution given is $435627650\mod ...
3
votes
1answer
231 views

The relationship between inner automorphisms, commutativity, normality, and conjugacy.

An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$ I have three somewhat broad questions about this: Why is it related to ...
4
votes
2answers
920 views

How to enumerate subgroups of each order of $S_4$ by hand

I would like to count subgroups of each order (2, 3, 4, 6, 8, 12) of $S_4$, and, hopefully, convince others that I counted them correctly. In order to do this by hand in the term exam, I need a ...
0
votes
2answers
59 views

When does $S_n$ have a subgroup with order $p^2$ where $p$ is prime?

I'm attempting this homework problem, and I'm not sure where to start. Here is the problem and how what I've got so far. Let $p$ be a prime number. What is the least positive integer $n$ such that ...
2
votes
0answers
37 views

Separable elements of a finite abelian group

Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively. ...
2
votes
2answers
89 views

$G$ $p$-group. If $H\triangleleft G$, then $H\cap C(G)\ne \{e\}$

I'm pretty new on this subject and I need a hint to begin to solve this question: If $G$ is a finite p-group, $H\triangleleft G$ and $H\ne \{e\}$, then $H\cap C(G)\ne \{e\}$ Thanks for any ...
8
votes
3answers
753 views

If $G$ has only 2 proper, non-trivial subgroups then $G$ is cyclic

Is the following true? If $G$ has two proper, non-trivial subgroups then $G$ is cyclic.
3
votes
1answer
67 views

Explicitly computing finite subgroups on elliptic curves

I have a simple cubic curve, say $y^2 = x^3 - x.$ Is there a simple way to find a small finite subgroup of points lying on this curve? (with respect to the elliptic curve group law.) Otherwise, does ...
4
votes
2answers
73 views

Finite Groups: $a \in G \implies a \in H$

Let $G$ be a finite group and let $H$ be a normal subgroup. Let $a$ be an element of G and suppose that $\gcd(|a|,[G : H]) = 1$. Show that $a$ is in $H$.
4
votes
1answer
91 views

Action of $S_4$ in $S_4/S_3$

Let $G = S_4$, $H = S_3$, $X = G/H$ be the set of right cosets of $H$, $x = (14)H$ and $G $ acts on $X$ by conjugation. Compute $\mathscr{O} (x)$ and $G_x$ (the stabilizer of $x$). I've got a ...
4
votes
2answers
256 views

Groups of order 8 are not simple

Show that any group of order 8 is not a simple group. I know that $\mathbb{Z}_8$, $\mathbb{Z}_2\times \mathbb{Z}_4$, $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$, $Q_8$, $D_4$ are not simple. ...
1
vote
2answers
125 views

If $H,K \leq G$ a finite group, then $\left\lvert HK \right\rvert = \cdots$ [duplicate]

If $H,K \leq G$ a finite group, then $$\left\lvert HK \right\rvert = \frac{\left\lvert H \right\rvert \cdot \left\lvert K \right\rvert}{\left\lvert H\cap K \right\rvert}.$$ The first part of ...
9
votes
0answers
152 views

Central Quotients of Finite Groups

There are more than 50 groups of order 48, and among them 16 groups have center of order 2, let $G$ be among such groups. Then $G/Z(G)$ is a group of order 24. What is this group of order 24? There ...
6
votes
1answer
203 views

Why is $PGL_2(5)\cong S_5$?

Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
7
votes
1answer
175 views

Non-central subgroup which is maximal among abelian normal subgroups is self-centralizing?

If $G$ is a supersolvabe group, and $A$ is a maximal among abelian normal subgroups of $G$, then the centralizer of $A$ in $G$ is $A$ itself (see link). My question is about the importance of the ...
5
votes
1answer
63 views

Automorphism of Graph $G^n$

I try to define the automorphism of $G^n$ where $G$ is a graph and $G^n = G \Box \ldots \Box G$,( $n$ times, $\Box$ is the graph product). I think that : $\text{Aut}(G^n)$ is $\text{Aut}(G) \wr S_n$ ...
0
votes
1answer
481 views

Prove the intersection of a Sylow $p$-subgroup and a subgroup is the unique Sylow $p$-subgroup

The statement we need to prove is: Let $P$ be a normal Sylow $p$-subgroup of $G$ and let $H$ be any subgroup of $G$. Prove that $P\bigcap H$ is the unique Sylow $p$-subgroup. Can you give some ...
11
votes
1answer
279 views

What groups can G/Z(G) be?

Let $G$ be a finite group and let $Z(G)$ denote its center. A simple result states that if $G/Z(G)$ a nontrivial cyclic group then $G$ is abelian. Of course if $G$ is abelian then $Z(G)=G$ and ...
8
votes
2answers
1k views

When is the group of units in $\mathbb{Z}_n$ cyclic?

Let $U_n$ denote the group of units in $\mathbb{Z}_n$ with multiplication modulo $n$. It is easy to show that this is a group. My question is how to characterize the $n$ for which it is cyclic. Since ...
1
vote
2answers
94 views

Conjugation in $S_n$

We have that any element in $S_n$ is generated by the adjacent transpositions $(12),\dots,(n-1,n)$. I am trying to calculate the conjugation of $(ab)\in S_n$ by $\sigma\in S_n$. So if we write sigma ...
2
votes
0answers
194 views

Element of a certain order in multiplicative group of residues

Let's say that $G$ is the multiplicative group of residues mod $p$, where $p$ is prime. I know that the order of an element $g \in G$ is the least $k$ for which $g^k \equiv 1 \mod{p}$. How can we go ...
4
votes
3answers
470 views

finite abelian group satisfying $x^2=e$

I looked but didn't see this question pop up. Not homework as I am graduating on Thursday and took Abstract a year ago. I'm taking the Praxis II and honing my skills. I have good intuition about ...
3
votes
2answers
2k views

A non-abelian group of order $ 6 $ is isomorphic to $ S_3 $

I know that it is duplicated. But I'm confusing some step of this proof. Please help me. pf) Let $ G $ be a nontrivial group of order $ 6 $. Since $ G $ is non-abelian, no elements in $ G $ have the ...
1
vote
1answer
27 views

Computing the number of elements of order $2$ and $3$, in the groups $L_{3}(q)$

What are the number of elements of order $2$ and $3$ in the groups $L_{3}(q)$? Also let $r$ be a divisor of $q^2+q+1$. What is the nuber of elements of order $r$ in the groups $L_{3}(q)$?
1
vote
3answers
132 views

If H is a p-group, the order of any H-orbit is a power of p.

This comes from the proof of the third Sylow Theorem in Michael Artin's "Algebra". Let S be the set of Sylow p-groups in a given group G of order $p^em$. Let H be any Sylow group. If we decompose S ...
0
votes
2answers
871 views

Filling up the Cayley table and finding Self-inverse

The set $G$ is given by $G = \{a, b, c, d, f, g, h, k\}$. $(G, *)$ is a group, with identity $k$, under a certain binary operation $*$. $a * b = c$, $b * a = d$, $f * f = a$, $g * g = b$, $h ...
2
votes
2answers
29 views

Ring $\mathbb{Z}/2mnr \mathbb{Z}$ unit, identity, orders

Let $p$ be a prime number which doesn't divide $2mnr$. So $p$ is a unit in the ring $\mathbb{Z}/2mnr \mathbb{Z}$ and $q=p^k$ for a certain $k \in \mathbb{Z}$ Could you explain to me why then: 1) ...
2
votes
1answer
49 views

Question about terminology, finite fields

My English is not very good, and that's why I would really appreciate it if you could explain to me what the phrase : these elements are under the same domain under $F$ and $\alpha$ means in this ...
2
votes
1answer
89 views

Fundamental counting principle for orbits going wrong?

maybe that's idiot, but I'm missing something here. Let $X = \{(123),(132),(124),(142),(134),(143),(234),(243) \}$, $A_4$ act on $X$ by conjugation (inner automorphisms) and $x = (123)$, then $4 ...
9
votes
2answers
397 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
4
votes
1answer
209 views

Dihedral group as a matrix group

I wish to consider the dihedral group as a matrix group. One way to do that is to consider it as a finite subgroup of $O_2$, a group of orthogonal $2\times 2$ matrices, defined by ...
1
vote
1answer
111 views

Let G be a finite group and let H and K be a subgroup of G so that [G:K] and |H| are relatively prime. Prove that [G:K]|H| divides |G|

So this is the proof: By Lagrange's Theorem we know $|G|=[G:K]|K|$ $|G|=[G:H]|H|$ Than we know that $|H|$ divides $|G|=[G:K]|K|$ Since $[G:K]$ and $|H|$ are relatively prime, than $|H|$ must ...
3
votes
1answer
250 views

Finite Abelian groups, G, H, K: $G \times H \cong G\times K$ then $H\cong K$

Let $G,H,$ and $K$ be finite abelian groups then if $G \times H \cong G\times K$ then $H\cong K$. I am trying to use the fundamental theorem for abelian groups to solve this, it is clear intuitively ...
6
votes
1answer
134 views

Groups with 20 Sylow subgroups

Is there a reasonably easy proof that a finite group with exactly 20 Sylow $p$-subgroups has PSL(2,19) or PGL(2,19) as a quotient group? What if we weaken this to merely: “a group of order 760 has a ...
4
votes
1answer
220 views

Metric on a group

Is there a non-abelian finite group $G$ with the property: If metric $d$ on $G$ is left invariant then is also right invariant?
0
votes
1answer
71 views

Metric in a group with order divisible by 3

Let $G$ be the non-abelian finite group whose order is divisible by $3$. Prove that exist a left invariant but not right invariant metric on $G$.
1
vote
3answers
265 views

An abelian group of order 6 has an element of order 6.

Let $ G $ be an abelian group of order 6. Then $ G $ has one element of order 6. And so, $ G $ is cyclic and isomorphic to $ \mathbb Z _6 $. In general, It is not true that an abelian group of ...