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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5
votes
2answers
620 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
78 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
121 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
351 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
1answer
833 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
25 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
108 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
594 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
28 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
48 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
75 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 ...
8
votes
2answers
283 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
137 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
84 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
204 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
114 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
188 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
67 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
134 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 ...
2
votes
0answers
65 views

the orbit of a root under operations of irreducible crystallographic group?

Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an ...
2
votes
0answers
187 views

$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$

Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
8
votes
1answer
495 views

Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
2
votes
0answers
82 views

Find the cycle index of cyclic group $C_p$ where $p$ is prime

Find the cycle index of cyclic group $C_p$ where $p$ is prime. No idea where to start...any help pointing me in the right direction would be appreciated.
3
votes
1answer
38 views

Different induced representations - same simples?

is the following case possible: $\pi_1, \pi_2$ two simple representations of the same subgroup over an arbitrary field. $\operatorname{Ind}(\pi_1)$ and $\operatorname{Ind}(\pi_2)$ have equal ...
24
votes
2answers
545 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
2
votes
1answer
145 views

Let $G$ be a simple group such that $|G|=p^2q r$ ($p$ and $q$ are distinct prime numbers) then how prove $|G|=60$?

Let $G$ be a simple group such that $|G|=p^2q r$ ($p$ and $q$ are distinct prime numbers, $r$ some positive integer) then how to prove $G$ and $A_5$ are isomorphic or ($|G|=60$)? Thanks in advance
1
vote
2answers
328 views

Is every group of order $21$ cyclic? [duplicate]

solution :- $21= 3 \times 7$ there is only one Sylow $3$ and Sylow $7$ subgroup so, Sylow $3$ and Sylow $7$ subgroup are normal in group $G$ so $G$ is cyclic group of order $21$. Am I right ? ...
-2
votes
3answers
179 views

Prove or disprove isomorphic graphs

We have no diagram, just a presentation. Symbols $\{a,b\}$. Rules: $(a^2 =1, b^2 =1, aba =bab)$. Everything is to be deduced from these rules only, here every word is equivalent to exactly one of ...
5
votes
1answer
344 views

Primitive permutation groups.

Suppose that G is a primitive permutation group of degree n which contains a 3–cycle. I want to show that G contains the alternating group. Here G being a primitive permutation group of degree n ...
7
votes
0answers
95 views

Clifford theory and induction

in the answer to this post there was the statement that a representation $\vartheta$ of a subgroup $\langle z\rangle$ can extend to a representation of the whole group $D_{2n}$. If I start the other ...
2
votes
2answers
113 views

Isomorphism of faithful representations

Let $G$ be a group and $f,g: G \rightarrow GL(V)$ be two faithful representations over some field $K$ with $f:x\mapsto f(x)$ and $g:x \mapsto f(x^{-1})$. I would like to find out if $f$ and $g$ are ...
3
votes
1answer
142 views

There exist Sylow subgroups $P$ and $Q$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.

From D&F's sylow theory section: Show that if $n_p\not\equiv 1 \mod p^2$ then there are distinct Sylow $p$-subgroups $P$ and $Q$ of $G$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$. Are ...
0
votes
1answer
60 views

Name the digraph of these transformation subsemigroups

I am trying to track down the name of this structure and some references. You take all members of the transformation semigroup on $n$ elements, $T_n$. For two members $x, y\in T_n$: if $x$ is in the ...
4
votes
3answers
381 views

Can a group of order $55$ have exactly $20$ elements of order $11$?

Can a group of order $55$ have exactly $20$ elements of order $11$? Give a reason for your answer by sylow theorem the answer is easy but without using sylow how can I solve this.can anyone help me ...
2
votes
3answers
115 views

Why is $| \rm{Aut}(\mathbb{Z}_n) | = \phi(n)$?

If $G$ is a finite cyclic group of order $n$, prove that $| \rm{Aut}(G) | = \phi(n)$, where $\phi(n)$ is the Euler's totient function. Can someone please help me with this?
4
votes
1answer
139 views

Cyclic Sylow $p$-subgroups are central in their normalizer when $p$ is the smallest prime divisor of $|G|$.

Let $p$ be the smallest prime dividing the order of a finite group $G$. If $P$ in $\operatorname{Syl}_p(G)$ and $P$ is cyclic, prove that $N_G(P)=C_G(P)$. This is not homework. It is from ...
16
votes
2answers
385 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
4
votes
0answers
120 views

Mackey criterion for normal subgroups

I am wondering how Mackey's criterion works for arbitrary fields. If there is a representation $\vartheta$ of a subgroup of $G$, then the induced representation $\operatorname{Ind}(\vartheta)$ is ...
2
votes
1answer
288 views

Question on groups of order $pq$

Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that If $q\mid p-1$ then there exists a non abelian group of order $pq$. Any two non-abelian groups of ...
1
vote
2answers
66 views

Different actions of an affine primitive group?

Fairly new to group actions and I'm having trouble finding answers to these in textbooks... Say we have a primitive action of $G$ on $\Omega$, with regular elementary abelian socle $N$. Now suppose ...
4
votes
4answers
101 views

Explain why $\newcommand{\Z}{\mathbb{Z}} U_{44} \cong (\Z_{10} \oplus \Z_2) $.

Explain why $\newcommand{\Z}{\mathbb{Z}} U_{44} \cong (\Z_{10} \oplus \Z_2) $ I know that $\Z_{20} \cong (\Z_{10} \oplus \Z_2)$, so if I can show $U_{44} \cong \Z_{20}$, then I can conclude that ...
0
votes
4answers
137 views

For which finite groups, number of elements of order $p$ is not $p-1$?

Let $G$ be a finite group and $p\mid |G|$ be prime. Can $G$ have exactly $p-1$ elements of order $p$? (except trivial groups which are isomorphic to $\Bbb Z_p$) I remember something similar to it. ...
0
votes
3answers
527 views

Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian [duplicate]

Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
7
votes
1answer
215 views

How do combinations (not permutations) relate to group theory?

First question. I'm just generally curious about combinations in group theory. How do they relate? If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
3
votes
0answers
70 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
1
vote
2answers
76 views

Show that every subgroup of $Q_8$ is normal.

Show that every subgroup of $Q_8$ is normal. Is there any sophisticated way to do this ? I mean without needing to calculate everything out.
7
votes
0answers
331 views

Why is the Monster group the largest sporadic finite simple group?

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
1
vote
5answers
71 views

Show that If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$

If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$. I am sort of stuck with this proof and I would appreciate a hint (not a full solution, please!). Preferably, ...
2
votes
2answers
128 views

How many subgroups of $\Bbb Z_5\times \Bbb Z_6$ are isomorphic to $\Bbb Z_5\times \Bbb Z_6$

I am trying to find the answer to the question in the title. The textbook's answer is only $\Bbb Z_5\times \Bbb Z_6$ itself. But i think like the following: Since 5 and 6 are relatively prime, $\Bbb ...
3
votes
2answers
778 views

Let $N$ be a normal subgroup of index $m$ in $G$. Prove that $a^m \in N$ for all $a \in G$.

I'm trying to understand this proof: Let $N$ be a normal subgroup of index $m$ in $G$. Prove that $a^m \in > N$ for all $a \in G$. Proof $\;\;$ Let $a\in G$. Since $[G:N]=m$, then $|G/N|=m$. ...