Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, and other advanced topics.

learn more… | top users | synonyms (1)

0
votes
0answers
8 views

Intersection of normal subgroups proof

Show that the intersection of normal subgroups is normal. Let $H_1$ and $H_2$ be normal in $G$, meaning $\forall a \in G$, $aH_1 = H_1a$ and $aH_2 = H_2a$. We show that $a (H_1 \cap H_2) = (H_1 ...
1
vote
0answers
5 views

How to show $\langle a^k\rangle\leq \langle a^\ell\rangle\Leftrightarrow \gcd(\ell, n)\mid \gcd(k, n).$?

Consider the cyclic group $G=\langle a\rangle$ where $o(a)=n$ where $o(a)$ means order of $a$. I'd like to show: $$\langle a^k\rangle\leq \langle a^\ell\rangle\Leftrightarrow \gcd(\ell, n)\mid \gcd(k, ...
1
vote
2answers
29 views

How to prove if $G$ is a group with every non-identity element having order 2 and $H$ is a subgroup, $G/H$ is isomorphic to a subgroup of $G$.

This isn't a homework problem. I'm preparing for an exam, and I have no idea how to solve this problem. Let $G$ is a group such that every non-identity element has order $2$. Let $H$ be a ...
0
votes
0answers
10 views

If $|G|=mp^e$ and $H\leq G$ then $|H|=n p^r$?

If $G$ is a finite group then I may write $|G|=m p^e$ with $p$ prime and $p\not \mid m$. Is it true that if $H\leq G$ then $|H|=n p^r$ with $p\not \mid n$ and $r\leq e$? If the statement above ...
4
votes
1answer
23 views

Integral domain ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$

Exercise: Show that the smallest subdomain of complex numbers containing the element $\alpha=\frac{\sqrt{5} - 1}{2}$ is ${\mathbf Z}[\alpha] = \{a + b\alpha\ |\ a,\ b \in {\mathbf Z}\}$. I thought I ...
0
votes
1answer
21 views

How to factorise a number in $\mathbb {Z}[\sqrt {-5}]$?

I am studying quadratic number fields. I have a question about factorization in $\mathbb {Z}[\sqrt {-5}]$ which seems less trivial than factorization in the Gaussian integers. Let $ w=\sqrt {-5} $. ...
-1
votes
3answers
19 views

Let $H$ and $K$ be subgroups of a finite group $G$ such that $|H|>\sqrt{|G|}$ and $|K|>\sqrt{|G|}$. Show that $|H\cap K|>1$.

Problem. $|H|>\sqrt{|G|}$ and $|K|>\sqrt{|G|}$ and $H,K\leq G$ where $G$ is a finite group. Prove $|H\cap K|>1$. $$|HK|=\dfrac{|H| |K|}{|H\cap K|}>\dfrac{|G|}{|H\cap K|}\Rightarrow |H\cap ...
1
vote
2answers
19 views

Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb Q[\alpha]|)=1$

Please, help me to understand this problem: Let $\alpha=\sqrt[3]{2}$ be a root of the polynomial $x^3-2$. a) Prove that the number of automorphisms in $\mathbb Q[\alpha]$ equals $1$ $(|Aut\mathbb ...
1
vote
0answers
17 views

Affine Buildings

I am trying to study affine buildings. So far I learn a lot of theoretical properties and definitions, but it was hard for me to find a good example of this object to "visualize" the theory. (Yes, I ...
0
votes
1answer
19 views

$H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,…,n)$ [on hold]

For $n>2$, if $H$ is the smallest subgroup of $S_n$ containing the transposition $(1,2)$ and the cycle $(1,2,...,n)$ , then (A) $H = S_n$ (B) $H$ is abelian (C) The index of $H$ in $S_n$ ...
0
votes
1answer
29 views

Let G be a group of order 17. What is the total no. of non-isomorphic subgroups of G?

I don't know how to find non-isomorphic subgroups of a group. Please explain in detail. Thanks.
2
votes
1answer
32 views

Group objects in category of $\mathcal{Set}$ are groups - How to proof?

Reading about group objects in categories, it's a fact that a group object is in the category of $\mathcal{Set}$ just a common group. I am trying to give an actual proof of this, but I'm a bit ...
-3
votes
0answers
21 views

How to list all ideals in some quotient ring of $GF(2)[x]$ [on hold]

List all the ideals in $GF(2)[x]/({x^{31}}-1)$ by their generator elements.
1
vote
1answer
29 views

Equivalent definitions of an algebra over a ring

I have seen two different definitions from several sources of an algebra over a ring. From Wikipedia: Let R be a commutative ring. An R-algebra is an R-Module A together with a binary operation ...
1
vote
3answers
35 views

Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$

Let $G$ be a group. Let $a$ be an element. Let $n,k$ be pozitive integers. Let $m$ be least common multiple of $n$ and $k$. Prove $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{m} ...
-1
votes
1answer
26 views

No sub-integral domain of Z with prime characteristic?

I try to find a subring of Z which it is integral domain and characteristic is a prime. Until now, I can't find it. But i believe that this proposition is true. Please help me prove or disprove.
0
votes
3answers
44 views

Give an example of an infinite non-commutative ring R with char(R)=15 [on hold]

I know I need to use matrices, but I'm not sure how. I know matrices are non-commutative however I'm confused about the characteristic part.
1
vote
1answer
24 views

Suppose that $L:K$ is algebraic. Show that the following are equivalent:

$(A)$ $L:K$ is normal $(B)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ then $j(L) \subseteq L$ $(C)$ if $j$ is any monomorphism from $L$ to $\overline{L}$ which fixes $K$ ...
2
votes
1answer
18 views

Proof verification for $fgh=1_A\dots\implies f,g,h$ are all bijections. - Cohn - Classic Algebra Page 15

Is the proof below correct? Thank you for your time! Notation: $xfgh\equiv h(g(f(x)))= (h \circ g \circ f)(x)$ Theorem: If $f:A\to B, g:B\to C, h:C\to A$ are three mappings such that $fgh=1_A$, ...
0
votes
1answer
29 views

General way to solve equations of congruent classes?

So I'm taking my first abstract algebra class, and I have some homework like "Solve the equation $x^2 + [3]\times x+[2]=[0]$ in $\mathbb{Z}_6$. This doesn't seem to difficult to do since I can easily ...
2
votes
2answers
44 views

Intuitive Meaning of Quotient Ring

I have been trying to understand the intuitive meaning of quotient ring for quite some times but unfortunately I could not find any. Now I am approaching this question from a different angle, hoping ...
1
vote
1answer
35 views

What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$?

Question: What does $[5^{2000}]$ equal to in $\mathbb{Z}_4$? My proof: (which I doubt whether its correct or not since it doesn't use the hint in the book) $[5^{2000}]=([5])^{2000}$ Since $5 \equiv ...
-2
votes
0answers
31 views

Galois group of $x^{2^k}+1$

What is the Galois group of $f(x)=x^{2^k}+1$ over $\mathbb{Q}$?
3
votes
1answer
28 views

The condition that $[LM:K]=[L:K][M:K]$ holds in the field.

Let $L$ and $M$ be intermediate fields of the extension $K\subset F$, of finite dimension over $K$. Assume that $L\cap M=K$ and $[L:K]$ or $[M:K]$ is 2, then $[LM:K]=[L:K][M:K]$ holds. How can I use ...
1
vote
2answers
40 views

Let D be an integral domain with characteristic 3. If x,y are elements of D then (x+y)^3 = x^3 +y^3

I am studying for an exam and the question is: Prove: Let D be an integral domain with characteristic $3$. If $x$ and $y$ are elements of $D$, then $(x+y)^3 = x^3 + y^3$.
-1
votes
1answer
26 views

If G is not commutative [on hold]

Edit: Since I did not provide enough detail in my explanation in OP: I have tried to prove this for the general case, but have not come across a suitable proof. I was unsure if I then needed to prove ...
1
vote
0answers
25 views

Showing that $\mathbb{Z}_3\times V\simeq\mathbb{Z}_2\times\mathbb{Z}_6$

I'm trying to show that $\mathbb{Z}_3\times V\simeq\mathbb{Z}_2\times\mathbb{Z}_6$, where $V$ is the Klein four-group, $\mathbb{Z}_2\times\mathbb{Z}_2$. I came up with the following method, which ...
1
vote
0answers
20 views

Prove polynomial f(x) irreducible over the integers.

Consider the polynomial $f(x) = (x − 1)(x − 2)···(x − (n - 1))(x − n) + 1$, for some $n \in \mathbb{N}$. Prove that $f(x)$ cannot be reduced to the form $f(x) = a(x) \cdot b(x)$ for polynomials $a, b$ ...
1
vote
1answer
34 views

Why in this sense this homomorphism is injective?

In this proof:enter link description here Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it ...
0
votes
3answers
80 views

Confused with Cayley's Theorem in group theory.

Cayley's Theorem: Every group is isomorphic to a group of permutations. $\mathbb Z_6$ is a group and $S_3$ is a permutation, but $\mathbb Z_6$ is not isomorphic to $S_3$. $\mathbb Z_6$ is ...
0
votes
1answer
22 views

Proving properties of a subgroup

Let $(G,\cdot)$ be a group, $H \subseteq G, H \neq \emptyset$. Let furthermore $X_{G,H}$ be defined as a construct with the following properties: $X_{G,H}$ is a subgroup of $G$ $X_{G,H} ...
0
votes
0answers
18 views

Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
1
vote
2answers
17 views

Showing that a set is a group, and a mapping is a group isomorphism

Let $(G,\cdot)$ be a group, $g \in G$. For $a,b \in G$ define $a * b := a \cdot g^{-1} b$. Show that $(G,*)$ is a group with the neutral element $g$ and $f : (G,*) \rightarrow (G,\cdot), a ...
1
vote
1answer
36 views

Show that it is a K-linear map

Let $K \leq K(a)$ a field extension with $[K(a):K]=n$. $K(a)$ is a vector space over $K$. How can I show that a map $\varphi : K(a) \rightarrow K(a)$, $\varphi(e)=ae$, is a K-linear map??
1
vote
1answer
24 views

quadratic field extensions of $\mathbb{Q}_p$

Today during class we proved that there were exactly three quadratic field extensions of the $p$-adic number field $\mathbb{Q}_p$. To prove this it was stated that it was enough to look at the group ...
0
votes
1answer
20 views

Find the minimum, irreducible polynomial

I have to find the minimum, irreducible polynomial of $$e^{\pi i/3}$$ over $\mathbb{Q}$. I have done the following: $e^{\pi i/3}$ is a root of the equation $x^6-1=0$, right?? ...
2
votes
1answer
46 views

hom(C) in category theory

I know in the basic definition of a category you have the class hom(C) of morphisms between objects in the category C. What never seems to be clear from textbook definitions is this: Are the members ...
1
vote
1answer
44 views

$f(x)$ is still irreducible

Let $f(x) \in K[x]$ an irreducible polynomial of $K[x]$ of degree $n$. Let $K\leq F$ a field extension with $[F:K]=m$. If $(n,m)=1$ show that $f(x)$ stays irreducible also as a polynomial of ...
1
vote
1answer
19 views

How to describe the quotient group Z x Z / < (4, -6)>

While solving a problem on group theory, I encountered the quotient group Z x Z / < (4, -6)>. Here Z is the integer. At first I thought it is just Z/(4Z) x Z/(6Z). But I was wrong. the quotient ...
-2
votes
0answers
33 views

Permutation Groups question in abstract algebra

Let G be a group and define a map $\lambda g : G \to G$ by $\lambda g(a) = ga$. Prove that $\lambda g$ is a permutation of $G$. Here, this is how I tried solving it. Kindly let me know if I am doing ...
0
votes
1answer
9 views

Help with finding cosets for cyclic subgroups

The question I'm working on is: Let $G=\mathbb{Z}_4\times\mathbb{Z}_3\times\mathbb{Z}_2$ and consider the subgroup $H=\langle\left(0,1,1\right)\rangle$ of G. Find all cosets of H. So I know that in ...
0
votes
0answers
14 views

Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
1
vote
0answers
7 views

The sum of pure submodules of noncommutative ring

Let $R$ be an arbitrary noncommutative ring, and let $A$ and $B$ be pure submodules of $R$. Is the sum $A+B$ a pure submodule of $R$? I feel it is not, but I could not find out a counterexample.
0
votes
1answer
17 views

Index of intersection of subgroups in group

Let $H$ and $K$ be finite index subgroups of a group $G$ with index $h$ and $k$, respectively. I know that $H\cap K$ is of finite index in $H$ and $K$. Is the index of $H\cap K$ in $H$ bounded by ...
4
votes
1answer
33 views

Number of different magmas up to isomorphism

Let $(M,\circ)$ be a magma over a finite set of order $n$. I tried to count all the possible magmas up to isomorphism, but I just can't get it right. My naive approach was to count all the possible ...
0
votes
2answers
24 views

Symmetries of a Polynomial

I was wondering how many symmetries the polynomial $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ has, and what they are. I got four: (i) $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ (ii) $(x_2-x_1)(x_1-x_3)(x_2-x_3)$ (iii) ...
1
vote
1answer
30 views

Any invertible linear map is homotopic to a composition of reflections

I am trying to solve a problem in Hatcher. I reduced my problem to showing that if $f:\mathbb{R}^n\to\mathbb{R}^n$ is an invertible linear map, then $f$ is homotopic to a composition of reflections, ...
4
votes
1answer
42 views

What can you say about $f$ and $g$ in the case that $fg$ is 1) Injective, 2) Surjective - Cohn - Classic Algebra Page 15

Question: Are my proofs below valid? In both cases we are using: $f:A\to B, g: B\to C$ Notation of your type converted: $(g\circ f)(x)=g(f(x))=xfg$ If $fg$ is injective what can be said about ...
0
votes
1answer
27 views

Finding isomorphism of a factor group based on orders.

If |G|=30 and |Z(G)|=5, what commonly known group is G/Z(G) isomorphic to?
0
votes
1answer
14 views

Primary decomposition of $Z_{1001}$ as a group of multiplication

The question is asking for the primary decomposition of $Z_{1001}$ as an abelian group under multiplication. So I did the following. By Euler $\phi$ function, I count the number of integers ...