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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4
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6answers
414 views

Abelian group of order 99 has a subgroup of order 9

Prove that an abelian group $G$ of order 99 has a subgroup of order 9. I have to prove this, without using Cauchy theorem. I know every basic fact about the order of a group. I've distinguished ...
1
vote
2answers
49 views

Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
1
vote
3answers
188 views

Two problems in Group theorem related to Sylow's theorem(maybe)

Prove that any subgroup of order $ p^{n-1} $ in a group $G$ of order $p^{n}$, p a prime number, is normal in $G$. $(a)$ Prove that a group of order 28 has a normal subgroup of order 7. To deal ...
4
votes
1answer
163 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
1
vote
0answers
31 views

show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ .

suppose that $G$ is finite group and $p$ is a prime number,then show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ where $q$ is arbitrary prime number and $q\neq p$ . ($O^{p}(G)$ ...
2
votes
1answer
76 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
4
votes
0answers
60 views

Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
3
votes
2answers
56 views

Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)

I am a beginner in group theory and I'm looking for finite groups that satisfy some properties. The only example I've found so far is: $$G_{q,c} = \{ f: \mathbb{F}_{q} \to \mathbb{F}_{q}, z \mapsto ...
0
votes
3answers
231 views

Group of order $p^2$ is commutative with prime $p$

Please help me on this one: Let $p$ be a prime number, show that each group of order $p^2$ is commutative. If you do not mind at all, could you please not give me the elegant explanation, but ...
0
votes
0answers
30 views

prove that $O^{P^{'}}(G)=P^{G}=P[P,G]$.

suppose that $G$ is finite group and $p$ is prime number.prove that if $P$ is a $p$-sylow subgroup of $G$ then $O^{p^{'}}(G)=P^{G}=P[P,G]$ which $P^{G}$ is normal closure of $P$ in $G$ . any hint or ...
3
votes
0answers
55 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
1
vote
2answers
115 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
9
votes
1answer
243 views

An abelian subgroup of symmetric group

PROBLEM: Let $G$ be an abelian subgroup of the symmetric group $S_n$ and $p_1, . . . , p_k$ be all prime divisors of $|G|$. Prove that $n≥p_1 +···+p_k$. QUESTION: How do you solve this problem. I've ...
7
votes
3answers
246 views

Normal subgroups in groups of odd order

I put the following question in my first-year algebra final this year: Suppose $G$ is a finite group of odd order and $N$ is a normal subgroup of order $5$. Show that $N\le Z(G)$. (By the way, this ...
1
vote
1answer
73 views

Groups of Order 12 aren't Simple

Suppose $G$ is a group of order $12=2^2*3$. Let $n_p$ denote the number of Sylow p subgroups. Then $n_2$ is 1 or 3 and $n_3$ is 1 or 4. I want to show that one of them is one since if that is the case ...
2
votes
1answer
102 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
3
votes
1answer
95 views

Constructing an irreducible representation for a finite group

This is not a homework. Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like: ...
3
votes
0answers
80 views

Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
1
vote
2answers
110 views

A question in finite group theory with proof using representation theory

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises in this field, but I have a problem with an exercise: I want to prove ...
2
votes
1answer
101 views

A question in character theory of finite groups

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises from "A Course in the Theory of Groups by Robinson" and "character ...
0
votes
1answer
61 views

Normalizer and Centralizer coincide

I am working on the following question: Suppose $G$ is a finite group that has a cyclic 2-Sylow subgroup $H$. I want to show that the centralizer, $C_G(H)$, and $\text{normalizer,} \ N_G(H)$ coincide. ...
5
votes
1answer
79 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
4
votes
2answers
772 views

How can I test if an element g is a generator of a group G with a known number of elements, N?

Let's say $G$ has $1000$ elements. Without looping through each $g^m$, how can I show that $g$ is a generator? I've deduced that I must prove that the order of $g = N$, or in this case $1000$, but I'm ...
1
vote
1answer
163 views

Non abelian group of order 40 [closed]

Let $G$ be a non abelian group of order 40 that is the direct product of two of its Sylow subgroups. Show that the center of $G$ has order 10. I tried 40=5×8 Since $G$ not abelian $G$ NOT EQUAL ...
0
votes
2answers
59 views

abstract algebra question with p-subgroup

Let $G$ be a finite group, $H<G$ a subgroup. Suppose that $H$ is a $p$-group, $p$ a prime, and $p\mid[G:H]$. Show that $p\mid[N_G(H):H]$ and $H<N_G(H)$. In particular, if $G$ is a $p$-group and ...
3
votes
4answers
478 views

Is it always true that $(a,b,c)(a,b,c) = (a,c,b)$?

I noticed that $(1,2,3)(1,2,3) = (1,3,2)$, and I also noticed that $(1,4,3)(1,4,3) = (1,3,4)$. Now, my question is whether or not it is true that for any permutation $(a,b,c)^2 = (a,c,b)$?
0
votes
1answer
131 views

Prove homomorphism and surjectivity of a function

I have a question about this exercise for my math study: Let $d, n \in\mathbb{Z}_{>0}$ with $d\mid n$. a) Prove that there is a homomorphism $g: (\mathbb{Z}/n\mathbb{Z})^*\rightarrow ...
3
votes
1answer
46 views

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular.

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular. I proved that if $N$ is intransitive, then $G$ will be imprimitive and Frobenius but I don't ...
2
votes
1answer
140 views

Generalized Cauchy's theorem (group theory)?

I think there is a corollary for Cauchy's theorem such that a finite group $G$ contains a subgroup of order $n$ for each divisor $n$ of $|G|$ with a certain condition (I can't remember what it was). ...
3
votes
1answer
157 views

Which of the following abelian groups are cyclic groups?

Given the abelian groups of order $7425$: $$Z_{33} \times Z_{15} \times Z_{15} , \ Z_{25} \times Z_{297} , \ Z_{45} \times Z_{165} , Z_{55}\times Z_9 \times Z_{15}$$ Which of these groups, if ...
0
votes
1answer
96 views

Number of Sylow subgroups

Prove that no group of order $56$ can be simple using steps ●finding sylows number 2-subgroups and sylow 7-subgroups ●explain why any of sylow 7-subgroups must intersect trivially, but this is not ...
3
votes
4answers
48 views

$G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$?

If $G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$ ?
-1
votes
1answer
81 views

Application of Chinese remainder theorem

from Chinese remainder theorem, we know that of $m,n \in \mathbf{Z}$, $(m,n) = 1$, then $Z_m \otimes Z_n \cong Z_{mm}$ as ring isomorphism, but how it related to the application that $\phi(nm) = ...
1
vote
0answers
73 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
1
vote
1answer
74 views

Evenness and oddness of group code weights

I'm doing exercises in Charles C. Pinter's book A Book of Abstract Algebra and I'm unable to solve problem 7 in section H of chapter 5 (subgroups). I think that there is a solution on this site but ...
1
vote
2answers
73 views

Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
3
votes
1answer
82 views

Finite abelian groups (application of structure theorem)

Problem Find all finite abelian groups that simultaneously have exactly $7$ elements of order $2$, exactly8 elements of order $3$, exactly $8$ elements of order $4$, at least an element of order ...
6
votes
1answer
261 views

Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it ...
3
votes
3answers
82 views

A group action proof without group actions?

I am currently teaching an undergraduate abstract algebra course out of Saracino, Abstract Algebra: A First Course. Exercise 13.13 asks the following: Let $K$ be the subgroup $\{ e, (1, 2)(3, 4), ...
0
votes
0answers
85 views

Sylow Theorems for Symmetric (Permutation) Groups

The General Linear Group $GL(n,\mathbb{F}_p)$ has an interesting property that the proof of Sylow theorem for this group can be given which is based on the natural action of this group on the ...
1
vote
1answer
66 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
1
vote
0answers
134 views

If $\sigma$ is a cycle of length $r$, then it has order $r$?

I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of ...
0
votes
3answers
76 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
0
votes
1answer
31 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
0
votes
2answers
55 views

Group theoretical proof of $\varphi(rs)=\varphi(r)\varphi(s)$ through generators of the group.

Given a group $G=\langle a\rangle$ of order $rs$, with $(r,s)=1$, I showed there exist unique $b,c\in G$ such that $a=bc$ with $b$ of order $r$ and $c$ of order $s$. The latter is a direct consecuense ...
2
votes
0answers
43 views

Characterize all $\mathbb Z[i]$ modules of order $21$ and $65$

I am trying to figure out how many $\mathbb Z[i]$ modules (up to isomorphisms) are of $21$ and $65$ elements. I've done a few similar exercises for finite abelian groups ($\mathbb Z$ modules) but I am ...
1
vote
5answers
422 views

How many ways can the vertices of an equilateral triangle be colored using three different colors?

Its not $3^3$ because some of the colorings are equivalent. How would I apply Burnside's theorem to this?
7
votes
1answer
309 views

endomorphism of finite groups

Have $\mathcal{G}$ denote the set of finite groups with at least $2$ elements. How would I go about showing that if $G \in \mathcal{G}$, then $\left|\text{End}(G)\right| \le ...
1
vote
1answer
52 views

Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
1
vote
0answers
66 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...