0
votes
1answer
8 views

Question about proof about index and subgroups

Let $G$ be a group so that $H\lhd G$. There is an element $g \in G$ so that $g$ isn't in $H$ but $g^2$ is in $H$. Show the index is even. Can't I just say that the cosets of $H$ are $H$ and $Hg$ ...
1
vote
3answers
18 views

A question about normal subgroups and index

Let $G$ be a group, and $H$ be a normal subgroup of $G$. $|H|=11$ and $[G:H]=24$. Let there be $x \in G$ and $x^{11}=e$. Show $x \in H$. Would like hints etc' on how to solve this. Is proving that ...
1
vote
1answer
17 views

Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
3
votes
6answers
44 views

Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of length one?

Wikipedia states that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to all complex numbers of length $1$. I have a hard time making sense out of this, and in particular, how complex numbers ...
1
vote
1answer
31 views

Let $G$ be a group. Then verify the statements with justification:

Let $G$ be a group. Then verify the statements with justification: $\bullet$ If $G$ has nontrivial centre $C$, then $G/C$ has trivial centre. $\bullet$ If $G$ does not equal $1$, there exists a ...
0
votes
1answer
37 views

Question about group theory and order in $\mathbb Z_n$

This is a only theoritical. Why is the order $o( \bar x )$ of $\bar x∈\mathbb Z_n$ the smallest non-negative integer $k$ such that $kx \equiv 0$ (mod $n$)? I don't understand how it follows from ...
3
votes
3answers
50 views

Only $1$ Nontrivial Subgroup $\Longrightarrow |G| = p^2$ [duplicate]

I am pretty new to this site , so I am not sure how things work, but I am in desperate help with a question that I don't know where to start or finish with. It is for a test I have to study for. Here ...
1
vote
1answer
30 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
2
votes
2answers
46 views

What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
1
vote
1answer
25 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
3
votes
1answer
35 views

How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
1
vote
1answer
21 views

Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
2
votes
1answer
41 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
2
votes
0answers
40 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
0
votes
2answers
54 views

Subgroup of group of order $44$

Pick the correct statement(s) below: $(a)$ There exists a group of order $44$ with a subgroup isomorphic to $ Z_2 + Z_2 $. $(b)$ There exists a group of order $44$ with a subgroup isomorphic to $ ...
0
votes
0answers
25 views

To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
2
votes
0answers
16 views

normal group $(\text{ker }g)(\text{ker } f)$ as a kernel of some group using $f$ and $g$

For group homomorphism $f: A\to B$ and $g: A\to C$ we know $\text{ker }f\cap \text{ker } g$ is kernel of $(f,g): A\to B\times C$. $(\text{ker }g)(\text{ker } f)$ is trivially normal in $A$, can we ...
1
vote
1answer
48 views

Is this type of a subgroup always normal? [duplicate]

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...
0
votes
1answer
48 views

If G is a finite group and $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$

Let G be a finite group. Show that, given $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$. I'm trying to use the info that is finite, but I can't find a way. For instance, if G is a ...
3
votes
2answers
75 views

What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
1
vote
2answers
63 views

An application of Sylow theorems in p-groups!

If $G$ is a finite group of order $p^{n}$ (which $p$ is a prime number) and have only one subgroup of order $p^{n-1}$ ,namely $H$ ,then $G$ is cyclic ! My "proof" is as follows: suppose $$x\in G-H$$ ...
1
vote
1answer
24 views

Lower bound of the index of a subgroup of a non abelian simple group

Let $G$ be a simple , non abelian group . Let $H$ be a subgroup of $G$ such that $[G:H] < \infty$ . Show that: $[G:H] \ge 5$
3
votes
1answer
17 views

Derived subgroup of the base group of a standard wreath product

Let $G=H\wr K$ be the standard wreath product with $K\ne 1$. Prove that $B'\leq [B,K]$ where $B$ is the base group of $G$. Deduce that $G/[B,K]\cong (H/H')\times K$. This is problem 1.6.20 from ...
2
votes
2answers
38 views

Can these two quotient groups be isomorphic?

Let $N$ and $M$ be two normal subgroups of a group $G$. Then we can show that the set $NM \colon= \{\ nm \ | \ n \in N, \ m \in M \ \}$ is a subgroup of $G$, that $M$ is normal in $NM$, and that $N ...
0
votes
2answers
35 views

Is every normal subgroup the kernel of some endomorphism?

Let $G$ be a group, and let $N$ be a normal subgroup of $G$. Then there is the canonical homomorphism $\phi$ of $G$ onto $G/N$ with kernel $N$. This homomorphism is defined as follows: $\phi(g) ...
0
votes
1answer
17 views

Questions on proof that transvections are conjugate in $GL(V)$.

I have difficulty following the proof that transvections are conjugates in $GL(V)$, and for $n \ge 3$ even in $SL(V)$. I give the necessary definitions and the proof, with the problematic parts ...
1
vote
1answer
63 views

If $xy=x^{-1}y^{-1}$, does this imply $x=x^{-1}$

This seems like a simple enough question, trying to show that if the title condition holds, that a group $G$ of which $x,y$ are elements, then $G$ is Abelian. $$xy=x^{-1}y^{-1}=(yx)^{-1}$$ From here I ...
2
votes
0answers
31 views

Groups with single conjugacy class of subgroups.

I wish to know all those groups in which there is single conjugacy class of subgroups of fixed order.For example, In finite cyclic groups and in Alternating group of degree 4, number of conjugacy ...
0
votes
1answer
34 views

Find all sub groups of order $4$ in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$ . Are they all cyclic?

Find all sub groups of order 4 in $\mathbf{Z}_4 \oplus \mathbf{Z}_4$. Solution : $\mathbf{Z}_4 =\{0,1,2,3\}$ $O(1) = O(3) = 4$, $O(0) = 1$, $O(2) = 2$ Hence, I found the subgroups of order 4 as ...
1
vote
2answers
101 views

A group with finitely many subgroups must be a finite group

Show that a group that has only a finite number of subgroups must be a finite group.(Fraleigh, A First Course in abstract Algebra-7th Edition,pg.67) I could not show properly so I need help. Thank ...
0
votes
2answers
23 views

Cartesian Product of Sets and the Direct Product of Groups

I'm having a bit of confusion. I've tried to search youtube and whatnot but I could not find any explanations. My book says the following: The Cartesian Product is denoted by: $$S_1\times ...
1
vote
2answers
41 views

Can a the field of fractions or quotient field F of an integral domain R be free over some set as an R-module?

We know any integral domain R when extended to a quotient field F, then F is free as an F-module on the set {1}. Can this field be free over some set as an R-module.
0
votes
1answer
30 views

Groups/Sets Notation Question

Simple question: But what does the sigma small Y mean, does it just represent a group? Also have seen this with numbers, and not quite sure what it means. Thanks
1
vote
0answers
25 views

Schur-Weyl duality from Double Commutant Theory

Let $V$ be a finite dim complex vector space. Then $V^{\otimes n}$ carries an action by $S_n$ by permuting factors $\sigma(\pi)(v_1\otimes...\otimes v_n)=v_{\pi^{-1}(1)}\otimes...\otimes ...
3
votes
3answers
33 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
2
votes
1answer
38 views

Finding Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$. The mistake in this method?

We need to find the Number of Cyclic Sub groups of order $15$ in $Z_{30} \bigoplus Z_{20}$ . This method does not give me the right answer (i.e $6$ ) . Attempt: We need to find the number of Cyclic ...
3
votes
1answer
58 views

A question about groups generated by two elements.

Suppose a group $G=\langle a,b \rangle$ and $|G|<\infty$ where $|a|=m_0$ and $|b| = m$. How is it that the operation table for $G$ can be completely determined just by knowing $ab=b^na$ for some ...
5
votes
4answers
96 views

Do we have $(G/H)\times H \cong G$ for groups in general?

After some thought I began to suspect $(G/H)\times H \cong G$, so I tried to construct an isomorphism by hand. I came up with $\varphi: (gH, h) \mapsto gh$ which came out to work provided $G$ is ...
1
vote
2answers
30 views

the smallest quasigroup, which is not a group

I'm wondering, which is the smallest quasigroup, which is not a group? And how to check it?
2
votes
1answer
50 views

Isomorphisms between finite abelian groups and cyclic groups

If G is abelian of order 175 and H is cyclic of order 25 and there is a homomorphism from G onto H then what is G isomorphic to? I can see how G is isomorphic to either $C_{25} * C_7$ or to $C_5 * ...
1
vote
1answer
59 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
27 views

Range and kernel of groups

Let $f: G \rightarrow H$ be a homomorphism. If the range of $f$ has $n$ elements, then $x^n \in$ ker $f$ for every $x \in G$. I can kind of understand why this is true. The ker of $f$ is $\{x \in ...
1
vote
1answer
38 views

Cyclic and abelian group

A group $G$ has order $25\cdot 47\cdot 17$. Is it cyclic and/or abelian? I know that a group of order $47$ or $17$ is cyclic, should I somehow use it?
0
votes
1answer
22 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
0answers
33 views

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. Condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle ?$

Let $G$ and $H$ be finite groups and $(g,h) \in G \bigoplus H$. State a necessary and sufficient condition for $\langle g,h \rangle = \langle g \rangle \bigoplus \langle h \rangle$ Attempt : Let $l = ...
2
votes
1answer
62 views

Introducing multiplication of cosets

So, i have encountered two ways to introduce the multiplication of cosets, and i want to understand exactly what is happening in each, specifically in light of the multiplication of cosets being ...
3
votes
2answers
70 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
6
votes
4answers
218 views

How is this subgroup normal?

Let $G$ be a group, and let $U$ be a subset of $G$. Let $\hat{U}$ be the smallest subgroup of $G$ containing $U$. Then $\hat{U}$ is the intersection of the collection of all the subgroups of $G$ ...
2
votes
1answer
31 views

Surjective Homomorphism Symmetric group

For $G=S_4$ i'm having a bit of trouble following the solution. For the blue underline I was wondering if there is a strategy for spotting this relatively quickly. For the green underline I ...
3
votes
0answers
39 views

If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic

(i) If $p>2$ is a prime number, Prove that $U(p^k) $ is cyclic (ii) Prove that $U(p^n) \thickapprox Z_{(p^n -p^{n-1}) }$ Solution : If I prove that if $U(p^k) $ is cyclic, then, we know that ...