-1
votes
0answers
20 views

$|G| \geq 3 $ is finite and has subgroup with index more than 4 then $G$ is not simple.

suppose $G$ is a finite group with more than three elements,prove that if $H$ will be a proper subgroup of $G$ with this property that $[G:H] \geq 4$ then $G$ is not simple. I found this question a ...
1
vote
1answer
55 views

Isomorphisms in finite abelian groups

Let G be an abelian group of order 175 (=5*5*7). Assume $x^5=e$ has at least seven solutions. What is G isomorphic to? I see and can show that G is isomorphic to the its Sylow subgroups (orders 7 and ...
0
votes
1answer
21 views

Normal subgroup $N$, subgroup $U$, then $UN/N = U/N$.

Let $G$ be a group and $N \unlhd G$ a normal subgroup, $U \le G$ some subgroup. Then I guess $U / N$ is always some group, and moreover $U / N = UN / N$, because $UN / N = \{ unN : u \in U, n \in N \} ...
2
votes
1answer
28 views

How does the base of a group determine the “sort” of the elements in the group

I'm trying to study groups in Mathematica, and I've asked a question on Mathematica.SE that perhaps only someone from Math.SE could answer. Related: How does ...
0
votes
1answer
29 views

$p$-subgroups conjugate iff $\cong$ to Sylow p-subgroups of some other groups?

Let $G$ be a finite group and $p$ a prime such that $p^\alpha$ divides $|G|$ and $p^{\alpha+1} \nmid |G|$. I know that Sylow $p$-subgroups of $G$ are conjugate to one another but if we have some ...
1
vote
3answers
63 views

Can we find some constraint about order of $xy$ in a group $G$?

Can we determine order of $xy$ in $G$ if we know order of $x$ and $y$ ? I know that answer is yes for abelian groups and I guess the answer is no for nonabelian case. That is why I am lookking for ...
0
votes
0answers
32 views

Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
0
votes
1answer
49 views

Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
1
vote
4answers
39 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
1
vote
2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
2
votes
2answers
77 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
4
votes
1answer
117 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
0
votes
0answers
22 views

Sylow subgroups and normalizers [duplicate]

Let $H$ be a Sylow 3-subgroup and $K$ a Sylow 5-subgroup of a finite group $G$. Suppose $|H|=3$ and $|K|=5$ and $N_G(K)$, has an element of order 3. Show that $N_G(H)$ has an element of order 5.
0
votes
1answer
37 views

Is this object a group?

$\begin{array}{ccccccccc} \times&e_0&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ e_0&I&e_1&e_2&e_3&e_4&e_5&e_6&e_7\\ ...
0
votes
1answer
15 views

What is the formula for products in dihedral groups?

Let $G \colon = \langle x, y \ | \ x^2 = y^n = e, \ x^iy^j = x^{i^{\prime}} y^{j^{\prime}} \ $ if and only if $ i = i^{\prime}, j = j^{\prime}, \ xy = y^{-1}x \rangle$. That is, let $G$ be the ...
1
vote
0answers
19 views

Let $G$ be a group of order 24 and suppose $n_2(G) > 1 \ \ and \ \ n_3(G) > 1$ . Then $G \cong S_4$

My attempt is : Since $n_3 > 1$ and $n_3 \equiv 1 \ \ mod \ \ 3 $ and divides 8, then the only possibilty is $n_3 = 4$ and thus $| G:N| = 4$, where $N = N_G(P)$ and $P \in Syl_3(G)$. Then $G/K $ ...
4
votes
2answers
101 views

Why free presentations?

What is the motivation to study "free" presentations of groups,even though all (or almost all) the questions (or the problems) concerning this type of presentations are known to be undecidable ?
1
vote
2answers
64 views

$g\in G$ maximal order in $G$ abelian then $G=\left<g\right>\oplus H$

If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$ Attempt: Using fundamental theorem I know that ...
0
votes
1answer
48 views

Subgroups of finite Abelian groups

I am interested in finding all of the subgroups (up to isomorphism) of a finite Abelian group $A$. I know the following: -- A finite Abelian group $A$ can be represented as a direct product of ...
3
votes
1answer
32 views

Having trouble grasping the class equation as an explanation as to why a conjugate class's order divides the order of a group.

Suppose $|G|$ is a prime power $p^n$ and that $N$ is a normal subgroup of $G$. Show that $|y^G|$ is a power of $p$ whenever $y \in G$ Attempt: Firstly, I assume that $y^G = \{ gyg^{-1} | g \in G ...
3
votes
1answer
41 views

Property of isomorphic subgroups in finite groups

I have the following question: Does there exist a finite group $G$ and two subgroups $U,H\leq G$, s.t. the following properties are satisfied: a) $H\cong U$. b) There is no subgroup $L$, s.t. ...
1
vote
1answer
29 views

$2$-Sylow subgroups of $S_{14}$

I need to describe what are the $2$-Sylow subgroups of $S_{14}$. I don't know how to start thinking about this since is it too large. I know that all that I must do is to find one of them since all ...
0
votes
1answer
14 views

Show that the orbit of $H$ containing $x$ is equal to the right coset $Hx$

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $X$ be the set of elements of $G$. Let $ \ast : H \times X \to X$ be given by $$ h \ast x = hx (h \in H, x \in X)$$. QUESTION: Let $x \in X$. ...
4
votes
2answers
87 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
1
vote
1answer
23 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
1
vote
2answers
31 views

Product of cyclic groups

How can you quickly tell that the product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_3$ has a 2-subgroup containing an element of order 4? Also, I don't understand the notion of multiplying ...
0
votes
1answer
42 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
1
vote
1answer
31 views

What about the index of this subgroup? [duplicate]

Let $G$ be a group, and let $H$ be a subgroup of finite index in $G$, and let $N \colon = \cap_{x \in G} \ xHx^{-1}$. Then $N$ is clearly a subgroup of $G$ which is contained in $H$ and such that ...
3
votes
3answers
78 views

Question about soluble and cyclic groups of order pq

I'm trying to solve the following problem: If $p>q$ are prime, show that a group $g$ of order $pq$ is soluble. If $q$ doesn't divide $p-1$, show that $pq$ is cyclic. Show that two non abelian ...
3
votes
0answers
45 views

How to solve this problem on finite groups? [duplicate]

Let $G$ be a finite group whose order is not divisible by $3$ and such that $(ab)^3 = a^3 b^3$ for all $a$, $b$ in $G$. Then can we determine if $G$ is abelian or not? Since $$ (ab)^3 = a^3 b^3 $$ ...
1
vote
1answer
73 views

Can someone please explain the word problem (from group theory) in Calculus III layman's terms

I'd appreciate it if someone could offer a schematic explanation of the word problem from group theory in terms which someone at a calculus III level (but without any background in group theory or ...
7
votes
6answers
195 views

The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
6
votes
1answer
117 views

Is $\langle a,b\; |\;a^7 = 1, ab = b^3a^3\rangle$ finite?

I've been playing a little with group definitions to see what kind of things I can make up. I'm struggling to prove that this group is finite. Can anyone point me in the right direction?
1
vote
2answers
56 views

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism on $g$, $g(N) =N$.

For a finite group $G$, if $N, G/N$ have relatively prime orders with $N$ normal, then for any automorphism $g$, $g(N) =N$. I can prove this for the case when there is a subgroup $H$ with the same ...
3
votes
1answer
40 views

Does G necessarily have a subgroup H…

I'm confused on an abstract math question. Let $G$ be a finite abelian group, and let $K$ be a subgroup of $G$. Does $G$ necessarily have a subgroup H such that $H≅G/K$ and $H∩K=⟨0⟩$. I think it is ...
0
votes
2answers
56 views

How to show that a group is finite and also normal

Let $G$ be an finite group and $H$ normal subgroup of $G$. Show $\left|G\big/H\right|=\left|G\right|$ if and only if $H=\{e\}$. Firstly I do not know how to show that $G$ is finite. Next I know that ...
12
votes
2answers
218 views

Can we uniquely determine a group given the orders of its elements?

Given a finite group $G$ and its order, consider a scenario in which we also know the orders of each of its elements. Does this information alone uniquely determine the group? If not, can we at ...
1
vote
1answer
37 views

A detail in Baer Theorem

I refer to Martin Isaac book, "Finite Group Theory", Theorem 2.12 pag. 55. In the proof there is a detail I can't understand. Our hypotesis are the following: $G$ finite group, $H\leq G$ s.t. ...
3
votes
1answer
108 views

Is the group $(G,*)$ abelian?

Let $(G,*)$ be a finite group in which the sets $C_a$={$x\epsilon G$|$ax=xa$} have the same cardinality, for all $a \epsilon G$ \{e}. My question is: is the group abelian?
1
vote
2answers
73 views

Using the Fundamental Theorem of Finite Abelian Groups

Let $G$ be a finite abelian group and let $p$ be a prime that divides the order of $G$. Use the Fundamental Theorem of Finite Abelian Groups to show that $G$ contains an element of order $p$. The ...
2
votes
3answers
94 views

For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?

Let $ (G,\ast)$ be a group with identity $e$ and cardinality $2n$ for some $n\in\omega$. Then, does there exist $x\in G$ such that $x\ast x=e$ and $x\neq e$?
1
vote
1answer
44 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
0
votes
0answers
20 views

Calculate the order of the generalized quaternion group and list its elements

Let $n \in \mathbb N$ and $w \in G_{2^n}$ a primitive $2^n$-th primitive root. Consider the matrices $R=\begin{pmatrix}\omega & 0\\0&\omega^{-1}\end{pmatrix}$ and ...
4
votes
2answers
99 views

Explicitly computing the isomorphism class of the tensor product of two finite abelian groups

How do I compute the isomorphism class of $A\otimes_\mathbb{Z} B$, where $A$ and $B$ are abelian of finite order? I can do this for a few examples, but I am unsure of how to proceed in the ...
3
votes
1answer
46 views

About product of $k$ commutators

I need help in folloving lemma. Lemma: Any product of $k$ commutators is expressible in the form $a_1^{-1}a_2^{-1}...a_{2k}^{-1}a_1a_2...a_{2k}$ I guess author means that any product ...
4
votes
2answers
124 views

Are there homomorphisms of group algebras that don't come from a group homomorphism?

Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces ...
1
vote
2answers
49 views

Groups and subgroups

I have been told that {0, 2, 4, 6, 8} is a subgroup of the multiplicative integer mod 10. I know that the operation is multiplication, so I understand that every element has its inverse within the ...
1
vote
1answer
68 views

Show that the p-Sylow subgroup is normal in $G$

Let $G$ be a finite group and suppose that $\phi$ is an automorphism of $G$ such that $\phi^3$ is the identity automorphism. Suppose further that $\phi(x) = x$ implies that $x = e$. Prove that for ...
1
vote
1answer
32 views

Prove $G$ has a normal Sylow subgroup

Let $|G|=pqr$ where $p, q$ and $r$ are prime and $p < q < r$. Prove that $G$ has a normal Sylow subgroup for either $p, q$ or $r$. Let $n_p, n_q, n_r$ denote the number of Sylow subgroups for ...
1
vote
0answers
40 views

irreducible polynomial of $\alpha$ over $\mathbb{Q}$

Let $\epsilon=\exp({\frac{2i\pi}{9}})$ and $\mathbb{Q}(\epsilon)$ be a cyclotomic extension of $\mathbb{Q}$ of order 9. If $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{Q}(\epsilon)$, determine the ...