0
votes
0answers
14 views

Prove that $U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$

If $m = n_1 n_2 \cdots n_k $ where $\gcd(n_i~,n_j)=1 ~~ \forall i \neq j$, then prove that: $$U(m) = U_{m/n_1} (m) \times U_{m/n_1} (m) \times \cdots\times U_{m/n_k}$$ where $\times$ refers to the ...
-1
votes
0answers
16 views

How to show that there are as many left cosets as there are right cosets?

G is a finite group and H is a subgroup, How to show that there are as many left cosets of H as there are right cosets?
1
vote
1answer
38 views

How to show that these two groups are isomorphic

I'm having trouble proving that two groups are isomorphic. I am having trouble with both the homomorphisms and the bijections. How would I go about solving this 2 part question: Prove that the ...
5
votes
5answers
96 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
-4
votes
0answers
33 views

Abstract Algebra: Prove that every field has only trivial ideals [on hold]

Prove that every field has only trivial ideals (that is, {0} and the field itself)
0
votes
0answers
37 views

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$

Let $G$ be a finite abelian group and let $p$ be a prime that divides order of $G$. then $G$ has an element of order $p$ Proof When $G$ is abelian. First note that if $|G|$ is prime, then $G \approx ...
0
votes
1answer
35 views

Classifying $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$

I wish to classify $\mathbb{Z}_{12} \times \mathbb{Z}_3 \times \mathbb{Z}_6/\langle(8,2,4)\rangle$ according to the fundamental theorem of finitely generated abelian groups. We have that it is of ...
2
votes
0answers
40 views

Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ such that $|S\cap T|$ is as larger as possibe and write $D = S \cap ...
0
votes
1answer
25 views

Ruling out orders when applying Sylow's theorems

Going through examples of applications of the Sylow theorems in Fraleigh's book, when proving that no group of order 36 is simple, after concluding that $| H \cap K | = 3$ for two $3$-Sylows $H$,$K$, ...
1
vote
2answers
33 views

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$.

Groups with $|G| = p^2q$. Prove that if $p$ and $q$ are primes, then there are no simple groups of order $p^2q$. Also another question, do p and q have to be distinct for this to hold? On top of ...
0
votes
2answers
35 views

Dihedralize Twice - dihedralize a dihedral group $D_n$

A simplest dihedral group $$D_4=C_2 \ltimes C_4$$ can be regarded as dihedralizing a $C_4$ by a semi-direct product. Q: Can one dihedralize the group $D_4$ a second time by defining $$C_2 \ltimes ...
0
votes
0answers
17 views

Unitary groups deals with matrices how it linked forms ??? [on hold]

But Projective Linear groups and all those things are defined on forms.. Is there any connection between forms and matrices... ??
1
vote
1answer
25 views

$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$

I have done part (i) I Was doing part (ii) and got stuck: Since from above I showed that $H_1 \times H_2 \subseteq K$ now i only need to show that $H_1 \times H_2 \supseteq K$. Let $(g_1,g_2) \in ...
-1
votes
0answers
21 views

Prove that G is a group acting on a set X. Where G= {(1),(123),(132),(45),(123)(45),(132)(45)} and X= {1,2,3,4,5}

I understand that the axioms that must be satisfied to prove that this is an "action" is: ex = x for all x an element of X (compatibility with identity). g_1(g_2*x) = (g_1g_2)*x (compatibility with ...
1
vote
1answer
33 views

Confusion with application of butterfly lemma in Lang's Algebra

In Lemma 3.3 of Serge Lang's Algebra, the so-called Butterfly Lemma is proved: And then Lang proceeds to prove Schreier's Refinement Theorem (highlight mine): In the proof of Schreier's theorem, ...
-8
votes
1answer
48 views

Abelian isomorphic group [duplicate]

Prove that if $G$ and $G'$ are isomorphic groups and $G$ is abelian, then $G'$ is abelian, too. Can you please solve this question, I have an exam soon and I have to learn this! I know I asked this ...
1
vote
3answers
44 views

Prove that if $G$ is a group and $H$ is a subgroup of $G$ generated by all elements of order $N$ in $G$, then $H$ is a normal subgroup of $G$.

I've tried proving that $ghg^{-1}\in H$ ($\forall g \in G$), but I don't see how the special property of $H$ guarantees this. Any insight? I've turned away from it to work on other things, and it's ...
1
vote
1answer
67 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
1
vote
2answers
26 views

Normal subgroup, quotient group, isomorphism.

Let $R^{*}$ be the group of nonzero real numbers under multiplication and let $R^{+}$ be the group of positive numbers under multiplication. Prove (a) $\{-1,1\}$ is a normal subgroup of $R^{*}$. (b) ...
7
votes
1answer
89 views

Show that if the prime $p$ divides $|G|$, then $|X|$ is divisible by $p$.

Question : Let $p$ be a prime number that divides the order of the finite group $G$. Let $X$ = $\bigcup_{P \in Syl_p(G)}P$. Show that $|X|$ is divisible by $p$.
2
votes
1answer
41 views

Cylic group of order 2

I need to check $\dfrac{D_{4}}{N}$ is isomorphic to the cylic group of order $2$. However, I just want to check if $\mathbb{Z}_2 $is the cyclic group of order 2 since it is not specified. Thank you.
1
vote
1answer
28 views

Order of the elements of a group of order $3$

Let $S$={$5$,$1/5$,$1$} be a set then I think $(S, .)$ is a group where identity element is $1$.Here order of the group is $3$.What is the order of the elements i.e $5$ and $1/5$? We know that the ...
1
vote
1answer
18 views

Automorphism of a group G

Let $G$ be a group, and let $a$ be a fixed element of $G$. Define the function $\gamma_a(x)=axa^{-1}$ for all $x \in G$. Prove that $\gamma_a$ is an automorphism. For $G=S_3$, compute $\gamma_{(1 \ 2 ...
0
votes
2answers
32 views

there is no subgroup of $\mathbb{Z}_4$ containing only 3 elements

Show that there is no subgroup of $\mathbb{Z}_4$ containing only 3 elements. I couldn't solve why 3 elements cannot exist. 0 and 2 are the only subgroup of $\mathbb{Z}_4$ with 2 elements. But 3 ...
0
votes
1answer
16 views

Show that $N_G(H) = H$ ( I.Martin.Isaacs)

Question : Let $P \in Syl_p(G)$, andsuppose that $N_G(P) \subseteq H \subseteq G$ , where $H$ is a subgroup . Prove that $H = N_G(H)$ we know that $H \subseteq N_G(H)$, please give me hint to show ...
2
votes
2answers
27 views

Finding all homomorphisms between $S_3$ and $\mathbb Z_8$

Finding all homomorphisms between $S_3$ and $\mathbb Z_8$ The method I saw on solving this was trying to figure out the homomorphisms based on the possible orders of image and kernel. Another thing ...
3
votes
2answers
87 views

If the intersection of a normal subgroup and the derived group is {e}, show that N is a subset of Z(G).

I think my reasoning is wrong, but if the intersection only contains the identity, doesn't that imply that the only commutator in N is {e}, so doesn't that mean N is automatically commutative? Why was ...
1
vote
1answer
32 views

Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$

Show that $Z_{p^2} \oplus Z_{p^2}$ has exactly one subgroup isomorphic to $Z_p \oplus Z_p$ Attempt: $Z_p \oplus Z_p$ has $p^2-1$ elements of order $p$ . Hence, all non trivial elements of $Z_p \oplus ...
0
votes
0answers
22 views

Can operations on arbitrarily differentiable functions form a group?

Consider the the set $S=${All uniary operations (all operations that perform on one function and give you another function, e.g $(x^2)'=2x$) on all arbitrarily differentiable functions}, does it form ...
0
votes
0answers
31 views

Show that $H<N_G(H)$ (I.Martin Isaacs)

Question : Let $H<G$ , where $G$ is a nilpotent group . Then $H<N_G(H)$ I know the proof of the result, I want to ask that if we take any group $G$ and its subgroup $H$ , then ...
0
votes
3answers
39 views

Problem with understanding homomorphism

Let $G$ be the group of all matrices of the form $\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix}$, $a,b,c \in \mathbb R$, ...
1
vote
2answers
53 views

Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$?

What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers. I've ...
2
votes
2answers
61 views

How do I prove that an order of a cycle is its length?

Let $\sigma$ be a cycle with length $n$ where $\sigma \in S_m$. How do i prove that $|\langle \sigma \rangle |$ is $n$?
1
vote
1answer
42 views

Prove the center of $G$ cannot have order $p^{n-1}$

Let $p$ be a prime, let $n>2$ be an integer, and let $G$ be a nonabelian group of order $p^n$. Prove the center of G cannot have order $p^{n-1}$. Honestly I have no idea where to start. Perhaps ...
0
votes
1answer
40 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
2
votes
1answer
31 views

Suppose that $G$ is a group of order 5 and let $\varphi: \mathbb{Z_{30}} \rightarrow G$ Determine the kernal of $\varphi$

Suppose that $G$ is a group of order 5 and let $\varphi: \mathbb{Z_{30}} \rightarrow G$ be and epimorphism. Determine the kernel of $\varphi$ Since $\ker\varphi \unlhd \mathbb{Z_{30}}$ (theorem) then ...
1
vote
1answer
23 views

silly confusion about subgroup proof: $< g,h> = \{g^r h^s : r, s \in Z\}$

Could someone help me understand this? It says certainly $<g,h> \supseteq \{g^r h^s : r, s \in Z\}$ which i understand. But now arent we suppose to show that $<g,h> \subseteq \{g^r h^s : ...
1
vote
0answers
28 views

proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian

If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian. Is that statement correct? Here's my attempt of proof: Let $a,b \in G$, then: ...
2
votes
2answers
25 views

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic.

Supose that G is a finite abelian group that does not contain a subgroup isomorphic to $Z_p \oplus Z_p$ for any prime $p$. Prove that G is cyclic. Attempt: If $G$ is a finite abelian group, then let ...
0
votes
2answers
23 views

Showing intersection of two finite-indexed groups is finite

Let $H, K$ be subgroups of $G$ with finite indexes, and $K\lhd G$, $H\lhd G$. Show $H \cap K$ has finite index. We were taught only first and second homomorphisms theorems, and not all the indexes ...
-3
votes
1answer
46 views

Showing to be nonabelian group

Show that $S_n$ is nonabelian group for $n≥3$. How can we show this? What are the conditions of being a nonabelian group? Thank you.
2
votes
1answer
36 views

Question about Quaternion group $Q_8$ and Dihedral group $D_8$

pretty much got stuck with the following question (it has several parts): a). Show that $D_8$ isn't isomorphic to $Q_8$ b). Let $K$ be a subgroup of $GL_2(\mathbb C)$ so that $$K=\left\langle ...
0
votes
1answer
25 views

Subgroups of $ D_8 $

$D_8 =\{ e, r, r^2, r^3, s, rs, r^2s, r^3s\} $ where r is a rotation by $ \frac{\pi } {2} $ anticlockwise and s is a reflection. Then according to my book, there are 10 subgroups, four of which are ...
0
votes
2answers
33 views

let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ , $n>1$. Prove that $G$ is not cyclic

let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a non trivial group and $n>1$. Prove that $G$ is not cyclic. Attempt : Let $G = G_1 \oplus G_2 ...
0
votes
2answers
37 views

Proof of a basic isomorphism.

How do you prove that $D_2 \cong V \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$? Where $V$ is the Klein-4 Group and $D_2$ is the dihedral group with cardinality 4. We have that $D_2 := \{1,r,s,sr \}$ ...
0
votes
1answer
22 views

tables of cyclic subgroups and conjugates

$G = S_5$, I need to construct tables for $H$ and $aHa^{-1}$ ($H =$ cyclic subgroup $(142)(35),$ and $a = (2354) \in G$) and see what can be inferred. In my attempt $H$ = $\{(142)(35), (124)(35), ...
-1
votes
1answer
28 views

Group Theory proving [on hold]

can someone help me with this question? 1) Given a natural number n≥1, let $G_n$ be the set of complex n-th roots of $1$, i.e. $G_{n} = \{z \in \mathbb{C} :z^n = 1\}$ Prove that $G_n$ is a group ...
0
votes
1answer
18 views

If $a\equiv b [p^k]$ then $a^p \equiv b^p [p^{k+1}]$

Can anyone explain the steps to this proof? I'm really lost/ If $k\geq 1$ and $a\equiv b[p^k]$ then $a^p \equiv b^p [p^{k+1}]$ Proof: Since $a= b + qp^k$ for some $q\in \mathbb{Z}$ we have $a^p = ...
6
votes
1answer
47 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
0
votes
3answers
56 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...