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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6
votes
2answers
308 views

Show that left cosets partition the group

I know how to prove that it happens, by proving that the left coset definition actually is an equivalence relation. Then, it's proved that it partitions the set, since equivalence relations do it. ...
1
vote
1answer
53 views

Finite (cardinality) modules over a PID

Let $R$ be a principal ideal domain with $|R|=\infty$. Suppose $M$ is an $R$-module such that $2 \le |M| < \infty$. What properties does this imply about $R$? Background: I was hoping that $R\cong ...
8
votes
2answers
92 views

Prove that $H$ is a normal subgroup of $G$

This is a problem from the book "Berkeley Problems in Mathematics": Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one coset of $H$ (...
1
vote
0answers
41 views

If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \unlhd G_{s} = G $ is a chief series of $ G $.

The $ p $-Fitting Subgroup of $ G $ is the maximal normal $ p $-nilpotent subgroup Of $ G $ and write it $ F_{p}(G) $. If $ G $ is a finite group. Suppose $ 1 = G_{0} \unlhd G_{1} \unlhd \cdots \...
1
vote
1answer
103 views

How to generate the icosahedral groups $I$ and $I_h$?

The icosahedral groups $I$ with 60 elements and $I_h = I \times Z_2$ are also three dimensional point groups. However, ever unlike other point groups, it seems there is rarely reference to give their ...
1
vote
1answer
62 views

How can I show that the characters in sense of irreducible representations are the same as the character maps from the burnside matrices?

My Task is: Let G be a finite group. 1. Let $C_1 = \{e\}, C_2,..., C_k$ be the conjugacy classes, and let $v_1,..., v_k$ be the normalised eigenvectors of the Burnside matrices of G, then for all s $\...
2
votes
1answer
73 views

Let $G$ be a group of finite order, $H$ and $K$ subgroups so that $H \unlhd G$; $K \unlhd HK \unlhd G$ and $(|H|,|K|) = 1$. Show that $K \unlhd G$.

I've been trying to solve this for a little while. I know that $H\cap K = \{1\}$ because of their orders and from the isomorphism theorems I know that $ HK /K \simeq H$. I've been trying to see if ...
1
vote
1answer
71 views

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $

Suppose $ G = G_{1}G_{2} $. For any prime $ p $, there exist $ P , P_{1} , P_{2} $ such that $ P = P_{1}P_{2} $, where $ P \in Syl_{p}(G) $ and $ P_{i} \in Syl_{p}(G_{i}) $ , $ i = 1,2 $. The proof ...
2
votes
4answers
68 views

Let $H, K$ be two subgroups of $G$. If $|H| = 12$ and $|K|=17$ then $H \cap K = \{e\}$

My reasoning: Since $|K| = 17$ and $17$ is prime, then any subgroup of $K$ is cyclic. Also, the order of any subgroup must divide the order of the group. But since the subgroups of $K$ must have an ...
2
votes
3answers
163 views

Proving that $gHg^{-1}$ is a subgroup of $G$

Let $G$ be a group and $H$ subgroup of $G$ with $\operatorname{ord}(H)=k$ I need to prove that $gHg^{-1}$ is a subgroup of $G$ $\color{grey}{(gHg^{-1}=\{ghg^{-1}\mid g\in G, h\in H\})}$ My ...
1
vote
3answers
254 views

$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p

Let $G$ be a non abelian group of order $p^3$, with $p$ prime. I'm proving that $Z(G)$ (its center) is of order $p$. I already know how to do it by saying that its order can't be $p^3$, nor 1, and if ...
3
votes
1answer
32 views

Finding subgroups of $\mathbb{Z}_{20}$

I need to find all the subgroups of $\mathbb{Z}_{20}$ My attempt: $\mathbb{Z}_{20}$ is cyclic $\Longrightarrow$ all the subgroups will be also cyclic, according to Lagrangh the order of the ...
0
votes
4answers
53 views

Calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$

I'm trying to calculate the group $\mathbb{Z}_m/2\mathbb{Z}_m$. I'm really bad with groups so I'd appreciate a verification of my conclusion: If $m$ is even then $\forall x\in \mathbb Z_m$ we get $2x\...
3
votes
1answer
62 views

Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$

I need to prove or disprove: Existence of isomorphism $\varphi:S_4\to \mathbb{Z}_8$ My attempt: No, there isn't isomorphism, because if it did then $S_4$ would have an element of order $8$, ...
0
votes
1answer
99 views

Number of elements of order $11$ in group of order $1331$

Let $G$ be a group of order $1331$. Prove that $G$ has at least $11$ elements of order $11$. $|G|=1331=11^3$ So by First Sylow's theorem, there exists a Sylow $11$-subgroup of G. By Third Sylow's ...
2
votes
2answers
88 views

Prove or disprove $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$

I need to prove or disprove: To the group $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$ My attempt: I just wrote all the details that I know: element in $A_5$ should be in form like $\...
12
votes
4answers
907 views

Product of all elements in finite group

Question: If $G$ is a finite group such that the product of its elements (each chosen only once) is always $1$, independent of the ordering in the product, what can we say about $G$? I was trying to ...
2
votes
0answers
53 views

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $

Let $ Q $ be a $ p$ -group and let $ N $ be a nontrivial, elementary abelian normal subgroup of $ Q $ which has a complement $ X $ in $ Q $. If $ Q = \langle y \rangle X $ for some element $ y $, then ...
3
votes
2answers
101 views

Finding order of $gag^{-1}$ in $G$ if $a^2=e\in G$

Let $G$ be a group, the order of $G$ is even, let $a \in G$, $a^2=e$ I need to find the order of $gag^{-1}$ in $G$ My attempt: $(gag^{-1})^2=(gag^{-1})(gag^{-1})=ga(g^{-1}g)ag^{-1}=ga(e)ag^{-1}=...
1
vote
1answer
78 views

Cardinality of automorphism groups of groups of order $p^4$.

As far as I know there is no classification of the automorphism groups of groups of order $p^4$. (see http://mathoverflow.net/questions/157049/classification-of-automorphism-groups-of-groups-of-order-...
2
votes
2answers
121 views

are these two finite groups with different presentation isomorphic?

Consider two groups $$\langle x,y \, | \, x^4=y^5=1 ,yxy=x \rangle$$ and $$ \langle a,b \, | \, a^{10}=1,b^2=a^5,aba=b \rangle.$$ I think they are isomorphic, but I can't show it, it will be great if ...
8
votes
3answers
652 views

Noncyclic Abelian Group of order 51

The problem is to prove or disprove that there is a noncyclic abelian group of order $51$. I don't think such a group exists. Here is a brief outline of my proof: Assume for a contradiction that ...
2
votes
2answers
88 views

Computing Factor Group

I am reading John Fraleigh's First Course in Abstract Algebra, $\S$36 on the Second Isomorphism Theorem which says that if $H < G$ and $N \triangleleft G$, then $$(HN)/N \cong H/(H \cap N).$$ He ...
3
votes
1answer
52 views

Find the order of $\tau^{100}$

Let $\tau= \left( \begin{array}{ccc} 1&2&3&4&5&6&7&8&9&10&11&12&13\\2&4&5&8&7&12&9&1&11&10&13&6&3\end{...
2
votes
2answers
76 views

Why normal subgroup chains in Galois theory

I have began understanding Galois theory and I had a question regarding the relationships of normal subgroups to field extensions. So given an irreducible polynomial over the rationals $$a_1 + a_2x +\...
3
votes
3answers
95 views

Show that $G/H\cong\mathbb{R}^*$

Let $G:= \bigg\{\left( \begin{array}{ccc} a & b \\ 0 & a \\ \end{array} \right)\mid a,b \in \mathbb{R},a\ne 0\bigg\}$ Let $H:= \bigg\{\left( \begin{array}{ccc} 1 & b \\ 0 & 1 \\ \...
2
votes
2answers
72 views

Finding subgroups of $\mathbb{Z}_{13}^*$

I need to find all nontrivial subgroups of $G:=\mathbb{Z}_{13}^*$ (with multiplication without zero) My attempt: $G$ is cyclic so the order of subgroup of $G$ must be $2,3,4,6$ Now to look for $g\...
4
votes
2answers
160 views

Is a non-trivial finite perfect group of order 4n?

A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$, or equivalently, if any $1$-dimensional complex representation is trivial. Question: Is a non-trivial finite perfect group of ...
2
votes
1answer
65 views

$G$ finite, $P < G$ a Sylow p-subgroup, $N_{G}(P)$ the normalizer is contained in $H < G$, show $N_{G}(H)=H$

Let $G$ be a finite group and $P<G$ be a Sylow p-subgroup. Let $N_{G}(P)$ be the normalizer of $P$ in $G$. Let $H<G$ be a subgroup containing $N_{G}(P)$. Prove that $N_{G}(H)=H$. I've been ...
6
votes
2answers
81 views

Name and role of a particular finite group?

The group generated by the functions $x\mapsto 1/x$ and $x\mapsto 1-x$ with composition of functions as the group operation is a non-abelian group with only six elements (listed below). Does this ...
2
votes
1answer
32 views

Group theory exercise from “Rational Points on Elliptic Curves”

Let $A$ be an abelian group and for every $m \geq 1$, let $A_m$ be the set of elements $P$ of $A$ such that $mP=0$. Now, suppose that $A$ has order $M^2$ and that for every integer $m$ dividing $M$, ...
3
votes
2answers
78 views

Subgroups of $\mathbb{Z}_{13}^*$ using “GAP language”

How can I know the nontrivial subgroups of $\mathbb{Z}_{13}^*$ (with multiplication without zero) using GAP language
2
votes
0answers
60 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
1
vote
1answer
56 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
1
vote
1answer
50 views

Latin square property sufficient?

So I know that for any group table, Every row must contain distinct group elements and the same holds for every column for a group table. And this property is called the Latin Square property. However,...
2
votes
1answer
118 views

Quantity of elements of order $d$ in $Z_n$, with $d \mid n $

I'm studying for an exam and I can't answer this problem. I'd appreciate a hint. What I've got so far: Let $x$ be an element or order $d$. Then, $x\cdot d \equiv 0 \pmod n \Rightarrow n \mid x\cdot ...
0
votes
0answers
47 views

Am I correct regarding Aut($Z_n$)

In the following pic- shouldn't it be $\Bbb{Z}_{{p_j}-1}$ instead of $\Bbb{Z}_{p_j}$. I think so because Aut$(Z_{p^n}) \cong Z_{p-1} \oplus \underbrace {(Z_p\oplus Z_p \oplus \dots Z_p)}_{n-1\ \...
3
votes
1answer
70 views

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple.

If $G$ is a group of order $2^nm$, where $m$ is odd and $(m-1)!<2^n$, show that $G$ is not simple. I started out by trying to prove this using the Sylow theorem, but it led nowhere. I was able to ...
2
votes
0answers
67 views

Upper bound on groups of order 60

I am aware of the fact that there are 13 non-isomorphic groups of order 60 but the proof of this is really long and something that I cannot present in a few minutes. Hence, I need to give a short ...
1
vote
0answers
70 views

Group isomorphism exercise

I am trying to solve the following problem: Let $G$ be a finite group such that there exists an isomorphism $\phi:f:G/Z(G) \to \mathbb Z_3 \oplus \mathbb Z_3$ and $x \in G$ such that $f(\overline{x})=...
3
votes
1answer
44 views

Why can we assume $N$ to be a $p$- group?

Let $G$ be a finite solvable group such that if three distinct primes $p,q$ and $r$ divides $|G|$ then $G$ does not contain any element of order the product of two primes and $G$ is minimal w.r.t ...
1
vote
1answer
80 views

Group of order $2m$ where $m$ odd has a subgroup of index 2. [duplicate]

Show that a group $G$ of order $2m$, where $m$ is odd, has a subgroup of index $2$. I am feeling a little dubious about my proof. Let $G$ act on itself by left multiplication to induce the ...
2
votes
1answer
34 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with $...
1
vote
3answers
139 views

Find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $.

I'd like to find all of the homomorphisms $ \varphi: S_3 \to \mathbb{Z}_{4} $. What I've tried so far: I tried to do $ \varphi(Id) = \bar{0} = \bar{4} $ (as somebody used here). But then I realized ...
2
votes
1answer
92 views

Finitely generated abelian group with certain properties

Problem Characterize all finitely generated abelian $G$ such that every proper subgroup of $G$ is cyclic, $G$ contains exactly two proper subgroups, and for each pair of subgroups $S$,$T$ in $G$ ...
0
votes
1answer
54 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
1
vote
0answers
49 views

Factorizations of Finite Abelian Groups

Every finite abelian group $G$ can be uniquely written as $$\mathbb{Z}/{d_1\mathbb{Z}} \times \mathbb{Z}/{d_2\mathbb{Z}} \times \cdots \times \mathbb{Z}/{d_r\mathbb{Z}},$$ where $d_i$ divides $d_{i+1}...
19
votes
1answer
264 views

Permutation of cosets

Let $G$ be a finite group and $\gamma \in Sym(G)$, such that $\gamma (1) = 1$ and $\gamma (gH) = \gamma (g)H$ for all $g\in G$, $H\leq G$. This means $\gamma$ induces a permutation of the left cosets ...
1
vote
0answers
47 views

Are there groups of order $p^4q^2$ which are not semi-direct product?

It is easy to show that if $G$ is a group of order $p^2q^2$, where $p,q$ are primes with correspondings Sylow subgroups $P,Q$, that $G$ is a semi-direct product of $P$ and $Q$. Moreover, if $pq\neq 6$,...
0
votes
1answer
35 views

Verify that $A \oplus B$, where $A$ and $B$ are cyclic groups of orders 2 and 3, is the cyclic group of order 6

Let's define $A$ and $B$ as follows: $A$ = {e,a} $B$ = {e,b,2b} Then $A\oplus B= \{\{e+e\},\{e+b\},\{e+2b\},\{a+e\},\{a+b\},\{a+2b\}\}$ which is equal to $\{\{e\},\{b\},\{2b\},\{a\},\{a+b\},\{a+2b\}\...