A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

learn more… | top users | synonyms (2)

1
vote
1answer
36 views

Is any $n$-parameter continuous group isomorphic to $\mathbb{R}^n$?

I know this is probably a dumb question. In a book I'm reading the following definition comes up. If $n$ is the minimum number of continuously varying real parameters required to characterize a ...
1
vote
2answers
77 views

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$. Ok, so now i know that $n_p(H) \le n_p(G)$ refers to the number of Sylow p-subgroups in H and G, respectively. From here, I ...
1
vote
1answer
40 views

do any two conjugates of [[1,1],[0,1]] generate SL(2,Z)?

Do any two (distinct) conjugates of the matrix $[[1,1],[0,1]]$ generate $SL(2,\mathbb{Z})$? Of course the conjugates $[[1,1],[0,1]]$ and $[[1,0],[1,1]]$ generate it, and some explicit computations ...
1
vote
1answer
52 views

Group of all $2\times2$ matrices where $a$, $b$, $c$, and $d$ are integers modulo $p$, Herstein Q$26$ Page $37$ [duplicate]

Let $G$ be group of all square matrices of order $2$ $$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$$ such that $a$, $b$, $c$, and $d$ are integers modulo a prime number $p$, such that ...
1
vote
1answer
77 views

Let $G$ be a group of order 35. Show that $G \cong Z_{35}$

Let $G$ be a group of order 35. Show that $G \cong Z_{35}$ Now, I am going to assume that this is a LaGrange based question. Also, I know that in order to be isomorphic, it must be one to one and ...
1
vote
1answer
46 views

Abelian Quotient group, which is not normal

What would be an example of a group $G$ with subgroup $H$ such that $G/H$ is abelian but $H$ is not normal?
1
vote
1answer
74 views

Prove that group $\mathbb{Q}\times Z_2$ is not isomorphic to $\mathbb{Q}$

I need some help with the following question. Prove that group $\mathbb{Q}\times Z_2$ is not isomorphic to $\mathbb{Q}.$ My proof: Let $a,b \in \mathbb{Q}$ and let $\phi$ be isomorphism.We have ...
1
vote
1answer
59 views

Prove that in any group $G$, we have $[G,G]\cap Z(G)\subseteq \operatorname{Frat}(G)$

Let $G$ be a group, we want to prove $[G,G]\cap Z(G)\subseteq \operatorname{Frat}(G)$. Can you please give some idea how to solve this? Here $\operatorname{Frat}(G)$ is the Frattini subgroup, ...
1
vote
2answers
71 views

Can we describe all group isomorphisms from $(\mathbb R ,+)$ to $(\mathbb R^+ , .)$ ?

Can we describe all group isomorphisms from $(\mathbb R ,+)$ to $(\mathbb R^+ , .)$ ? I have tried that if $f$ is such an isomorphism , then $f(x)>0$ , and $f(r)=(f(1))^r , \forall r \in \mathbb Q$ ...
1
vote
2answers
47 views

Exercise from Lang's Algebra concerning normal subgroups of relatively prime order.

Chapter 1, Exercise 13(a) Let $G$ be a group, and $H,N$ normal subgroups whose orders are relatively prime. Show that $xy=yx$ for all $x\in H$ and $y\in N$, and that $H\times N\cong HN$. Clearly, ...
1
vote
2answers
28 views

Let H and N be normal subgroups of a group G with H $\cap$ N = {e}. Prove that hn = nh for all h $\in$ H and n $\in$ N.

What does H $\cap$ N = {e} imply here? Is it related to the divisibility of the orders of H and N? If so, how do I relate that to show commutativity? Thanks in advance.
1
vote
2answers
49 views

Can we find the generator of the Galois group of $x^{p-1}+x^{p-2}+…1$?

$p$ is a prime. We know that $x^{p-1}+x^{p-2}+...1$ is irreducible in Q[x]. And the splitting field of $x^{p-1}+x^{p-2}+...1$ over $Q[x]$ is $Q(\xi_p)$-the primitive pth root of unity. Now I want to ...
1
vote
1answer
40 views

Prove that every finite subgroup of SL2 is subgroup of one of the following groups

I need to prove that every finite subgroup of $SL_2(\mathbb{Q})$ is a subgroup of one of the following groups: $D_3, D_4, D_6$. Let $G$ is a finite subgroup of $SL_2(\mathbb{Q})$, $ g\in G$. I can ...
1
vote
1answer
38 views

If $t=xy$ then how to prove that $tx$=$xt^{-1}$

Let $x$ and $y$ be elements of order $2$ in any group $G$.If $t$=$x$$y$ then how to prove that $t$$x$=$x$$t^{-1}$. I know that in Dihedral Group of order $2n$ if $s$ be a refection across a line ...
1
vote
1answer
27 views

A generating set of a finitely generated group

A group is called finitely generated if it has a presentation with finite generators. Edit: My original question was vacuous. Suppose that $G$ is a finitely generated group and $\{g_i\}_{i\in I}$ is ...
1
vote
2answers
67 views

Is $H$ a subgroup of $G$?

Let $G$ be a group and $a$ belongs to G. Will $H = \{a^{2n}:n\in Z\}$ be a subgroup of $G$? Won't $G$ have to be commutative?
1
vote
3answers
61 views

$G/Z(G)$ is cyclic then is abelian?

my question is if $G/Z(G)$ is order of p then is commutative then is abelian group but if G is abelian then $G=Z(G)$ therefore $G/Z(G)$ is not order of p?Is it contradiction?
1
vote
2answers
31 views

If $o(a),o(b)\gt 1$ and $o(a)$ and $o(b)$ are co-prime then $o(a)o(b)$ divides $|G|$

I'm trying to prove the following statement: Let $G$ be a finite group and $a,b\in G$ such that $o(a),o(b)\gt 1$. Prove that if $o(a)$ and $o(b)$ are co-prime then $o(a)o(b)$ divides $|G|$. Well, ...
1
vote
3answers
40 views

Multiplying two matrices that are the same size?

I've got this homework question for Group Theory and it states: $A=\begin{bmatrix} 1 \ 2 \ 3 \ 4 \ 5 \\ 1 \ 4 \ 3 \ 2 \ 5 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 ...
1
vote
1answer
59 views

Show that a group that has only a finite number of subgroups must be a finite group

Show that a group that has only a finite number of subgroups must be a finite group. I started by assuming that group is infinite. But, don't understand how I should go from this assumption
1
vote
2answers
66 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
1
vote
3answers
51 views

Building a proper homomorphism between groups.

Suppose I have a cyclic group $G$ of order $6$. I want to show that it is isomorphic to $\Bbb {Z}_6$. So $G=\{e,g^2,g^3,g^4,g^5\}=\langle g\rangle$. Can I build a homomorphism $f:G \to \Bbb{Z}_6$ that ...
1
vote
2answers
69 views

If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime?

The statement I'm trying to understand is as written in the title : If $ H , K$ are subgroups of $G$ and $HK$ is subgroup of $G$ then $|H|$, $|K|$ aren't co-prime? I tried to find a counter ...
1
vote
3answers
39 views

Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
1
vote
2answers
35 views

Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) [duplicate]

Find Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) I think we need to find order of centralizer of given permutation but how to find it?
1
vote
2answers
93 views

Direct product of simple groups

Let $G=H_1\times H_2$, $H_1,H_2$ are simple groups. Let $L\vartriangleleft G$ ($L$ isn't trivial). Show that $L$ isomorphic to $H_1$ or $H_2$. I tried to construct "projections" of $L$ on $H_1,H_2$, ...
1
vote
2answers
39 views

A question about the orders of the elements of a group [duplicate]

Let $m$ and $n$ be to positive integers strictly larger than $1$. Is it possible to find a group $G$ in which there are two elements, say $a$ and $b$, such that the order of $a$ is $m$, the order of ...
1
vote
3answers
103 views

Prove $\operatorname{Ker}(\phi)\ne \{e\}$

$G_1$ and $G_2$ are finite groups such that $|G_1|>|G_2|$ and $|G_2|$ divides $|G_1|$, Let $\phi\colon G_1\rightarrow G_2\space$ be a homomorphism such that $\operatorname{Ker}(\phi)\ne G_1$ Can I ...
1
vote
2answers
51 views

Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.

Suppose $[\mathbb{Z}_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}_p = n\mathbb{Z}_p \subseteq G$, after which I am done, since for any $x\in G$ we ...
1
vote
2answers
31 views

$H\unlhd G$ iff there exists a group $K$ and a homomorphism $\varphi:G\rightarrow K$ with $H=\ker(\varphi)$

Let $H\subseteq G$ be a non-empty subset. Then $H\unlhd G$ iff there exists a group $K$ and a homomorphism $\varphi:G\rightarrow K$ with $H=\ker(\varphi)$ I am confused as to what I have to show. Do ...
1
vote
1answer
64 views

Characteristic subgroups of a direct product of groups

Let $G=H\times K$ and $H\times 1$ be a characteristic subgroup of $G$. Then can we conclude that $1\times K$ is also a characteristic subgroup of $G$? My motivation is the case where orders of ...
1
vote
2answers
44 views

Nonabelian group of order $2p, p>2$ has trivial center

I'm stuck on an old algebra prelim problem. The problem is to prove that a nonabelian group of order $2p,p \text{ an odd prime}$ has trivial center. One thing I know is that any nonabelian group of ...
1
vote
2answers
63 views

Group Transitive Action's Effect on Stabilizers's Conjugacy

I am looking for guidance for two problems on group action, one of them is here and the other one has just been posted earlier: Assume that $G$ operates on a set $\Omega.$ Show that, if $G$ acts ...
1
vote
3answers
39 views

Intersect of Stabilizers is a Normal Group

I am looking for guidance for two problems on group action, one of them is here and the other one will be posted in another page: Assume that $G$ operates on a set $\Omega.$ Show that ...
1
vote
2answers
81 views

Groups such that inclusion on collection of all its subgroup is a total order

Characterize the Groups with the following property: Suppose G is any group such that for any two subgroups, H and K either H $\subseteq$K or K $\subseteq$ H. Now what can we tell about cardinality, ...
1
vote
1answer
91 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 ...
1
vote
1answer
50 views

Cayley Table Question

How can I figure out the order of the elements in a Cayley table?
1
vote
2answers
47 views

Determing whether (S, *) is an Abelian group

Given a set defined as $S=\{(a,b) | a,b \in \mathbb{Q} \land a^2+b^2=1 \}$ and a binary operation $*$ defined as $(\forall(a,b),(c,d) \in S) ((a,b)*(c,d) = (ac-bd, bc+ad))$, determine whether $(S,*)$ ...
1
vote
1answer
59 views

If $H \leq G, \exists g \in G$ such that $HgHg^{-1} = G$, then $H = G$

Just wanted some overall feedback from a homework question. Let $G$ be a group where $H \leq G$. Prove that if $\exists g \in G$ such that $HgHg^{-1} = G$, then $H = G$. $\it{Proof.}$ Note that $H = ...
1
vote
1answer
27 views

Showing that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$ for finite p-groups with order $|G|=p^3$

I have a finite, non-abelian $p$-group $G$ with $|G|=p^3$. I want to show that $|Z(G)| = p$ and $G/Z(G) \simeq \mathbb{Z}_p \times \mathbb{Z}_p$, where $Z(G)$ is the center of $G$. From the ...
1
vote
3answers
55 views

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$?

If $\gcd(a,b) = 1,$ then why is the set of invertible elements of $\mathbb Z_{ab}$ isomorphic to that of $\mathbb Z_a\times \mathbb Z_b$? I know the proof that as rings, $\mathbb Z_{ab}$ is congruent ...
1
vote
1answer
43 views

Proof of the existence of inverse elements for a group

The group to be determined is defined as follows: $\{x\in\Bbb{Z^4}:x_1x_4=1+x_2x_3\}$ with $(x,y)\mapsto(x_1y_1+x_2y_3,x_1y_2+x_2y_4,x_3y_1+x_4y_3,x_3y_2+x_4y_4)$ $*$ denotes the operation. We have ...
1
vote
2answers
33 views

Examples of group homomorphisms with isomorphic but not equal images

This may be a poor question. I am having trouble thinking of a pair of group homomorphisms: $\varphi, \Psi: G \rightarrow H$ between groups where $\varphi(G) \neq \Psi(G)$ but $\varphi(G) \cong ...
1
vote
2answers
47 views

Aut$(K/F)$ permutes roots of polynomial.

Let Aut$(K/F)$ is the set of all automorphism from $F$ to $K$, where $K$ is a galois extension of $F$. Let $f(x) \in F[x]$ and $\alpha$ be a root of the polynomial $f(x)$. I am able to prove that for ...
1
vote
3answers
39 views

division with a remainder

I have Problems with the prove of these exercises of my mathematics study. For (a), I tried to use long division to find the remainder of (2^a) - 1 and (2^b) - 1, but that didn't really work. Can you ...
1
vote
2answers
223 views

Let $G$ be a group and $H$ a normal subgroup of $G$. Prove: $x^2 \in H$ for every $x \in G$ iff every element of $G/H$ is its own inverse.

Let $G$ be a group and $H$ a normal subgroup of $G$. Prove: $x^2 \in H$ for every $x \in G$ iff every element of $G/H$ is its own inverse. Here is my proof. I've only tried proving one way ...
1
vote
2answers
63 views

Check identity in Gap

How can I compute the identity of commutators? For instance, Let $H$ be a group. Then I want to know about $[h_1,h_2] [[h_1,h_2],k_1] [k_1,h_1h_2]$.
1
vote
2answers
65 views

$NH=KN\;$ and $\;H\cap N=K \cap N$ imply $H=K$

Let be $G$ a group and $H$ and $K$ two subgroups such that $H\leq K \leq G$. Let be $N\trianglelefteq G$. How can I prove that the relationions $NH=KN\;$ and $\;H\cap N=K \cap N$ imply $H=K$? ...
1
vote
2answers
58 views

Fundamental group of countably many holes and one limit point

So I proposed this Fundamental group of a space of infinite genus and an accumulation point same question to my professor in Complex Analysis when we were going over the fundamental group and ...
1
vote
4answers
104 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...