Abstract algebra is a study of algebraic objects. Some of the more common algebraic objects are Groups, Rings, Fields, Vector spaces, Modules, and other advanced topics.
-3
votes
1answer
45 views
Finding a primitive root modulo $13$
Find a primitive root modulo each of the following integers.
a) $13$
My TA said we are not going to go over this. We did not go over the topic. It seems like something good to know though.
...
0
votes
1answer
41 views
Finding a primitive root modulo $11^2$
Find a primitive root modulo each of the following moduli:
a) $11^2$
My TA said he is not going to go over this so do not worry about it. He said you can try this if you want but he would ...
7
votes
1answer
62 views
If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?
If order of the element $a^5$ is 12 can we make any guess about the order of the element $a$ in a group $G$?
Could anybody clear my this doubt?
Thanks for the help
0
votes
1answer
34 views
$\bigcup_{x \in G} xHx^{-1} \neq G$
I could not solve it properly: $\bigcup_{x \in G} xHx^{-1} \neq G$ if $G$ is a finite group and $H$ is a proper subgroup (of $G$).
I tried to use the class equation and to create other actions to ...
2
votes
1answer
31 views
Action of $S_4$ in $S_4/S_3$
Let $G = S_4$, $H = S_3$, $X = G/H$ be the set of right cosets of $H$, $x = (14)H$ and $G $ acts on $X$ by conjugation. Compute $\mathscr{O} (x)$ and $G_x$ (the stabilizer of $x$).
I've got a ...
1
vote
2answers
35 views
There is an automorphism of $\mathbb Z_6$ which is not an inner automorphism
I'm trying to show that there is an automorphism of $\mathbb Z_6$ which is not an inner automorphism. Since the generators of $\mathbb Z_6$ are 1 and 5, then we have two choices, we exclude the ...
3
votes
2answers
91 views
Proper subgroup of $\mathbb{Q}^{+}$ with finite index
Is there a non-trivial subgroup $H$ of $\mathbb{Q^{+}}$, such that $|\mathbb{Q^{+}} : H|$ is finite? Of course, $|H| = \aleph_0$, but I could not prove that such $H$ does not exist (I think it does ...
6
votes
2answers
59 views
Finding elements of order $8$ in $\mathbb{Z}_{8000000}$.
I want to find elements of order $8$ in $\mathbb{Z}_{8000000}$.
I know that elements of order $n$ in a group $\mathbb{Z}_m$ is $\phi(n)$. But how can I apply this result for a group having such a ...
3
votes
3answers
85 views
Are Clifford groups very *non-commutative*?
Clifford groups seem to be very non-commutative by the relation \begin{equation}
\gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}.
\end{equation} But is it really so? Can we put this degree of ...
2
votes
1answer
52 views
Group of order 8 is not a simple group
Show that any Group of order 8 is not a simple group.
I know that $\mathbb{Z}_8$,$\mathbb{Z}_2\times \mathbb{Z}_4$ ,$\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$,$Q_8$,$D_4$ are not simple. ...
1
vote
1answer
35 views
Categorical Confusion in Rotman's Advanced Modern Algebra Second Edition
In Rotman's Advanced Modern Algebra, exercise 6.45 (ii) in the second edition, he gives us objects $X, C_1, C_2$ and morphisms $g_1: X \rightarrow C_1$, $g_2: X \rightarrow C_2$, and asks us to prove ...
3
votes
2answers
52 views
Morphisms in a category with products
I'm having a hard time proving that
$$(\psi\phi)\times(\psi\phi)=(\psi\times\psi)(\phi\times\phi),$$
where $\phi:G\to H$ and $\psi:H\to K$ in some category with products. I have seen a diagram of this ...
1
vote
2answers
27 views
Confusing with the concept of normalizer $N_G(H)$
I'm Confusing with the concept of normalizer $N_G(H)$.
It's a stupid question, sorry I'm new in this subject.
Following the Hungerford's concept:
If $H$ acts by conjugation on the set $S$ of all ...
1
vote
0answers
33 views
Full Rank Matrix with a specific construction
Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$
$$Z=\begin{bmatrix}
w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
1
vote
1answer
35 views
how to show associativity of multiplication for not just 3 operands but for n operands
ie Id like to show
a(bc)=(ab)c
but for any n operands
eg
abcdefg=gfdcabe etc
I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
-1
votes
4answers
41 views
Allowing the zero element in a field to have an inverse
In the definition of a field one of the required properties is that
every element other than zero has a multiplicative inverse.
It's vague whether the zero is forced not to have an inverse or not, ...
1
vote
2answers
37 views
If $H,K \leq G$ a finite group, then $\left\lvert HK \right\rvert = \cdots$ [duplicate]
If $H,K \leq G$ a finite group, then
$$\left\lvert HK \right\rvert = \frac{\left\lvert H \right\rvert \cdot \left\lvert K \right\rvert}{\left\lvert H\cap K \right\rvert}.$$
The first part of ...
-5
votes
0answers
55 views
Midterm Results: Linear Algebra (Retrospective Review) [closed]
$\diamond$ How does my professor know that $V'\subset N(T+S)$ suffices to show $\Omega$ is a subspace of ${\cal{L}}(V,W)$?
Find all real numbers $a\in \mathbb{R}$ such that the set ...
1
vote
2answers
31 views
Direct Products
If $A, B, C, D$ are groups and $B\cong C \times D$. Show: $A \times B\cong A \times C \times D$
What map should I use which will help me use the given isomorphism? Do I have to define two maps?
...
4
votes
2answers
68 views
Quadratic subfield of cyclotomic field
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
2
votes
2answers
49 views
Injective hull of $\mathbb{Z} _p$
The $p$-primary component of the group $\mathbb{Q}/\mathbb{Z}$ is denoted by $\mathbb{Z}(p^{\infty})$, where $p$ is a prime.
Now I want to show that
$\mathbb{Z}(p^{\infty})$ is an injective ...
0
votes
2answers
39 views
If $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$.
I'm trying to show that if $N$ is a normal subgroup of $G$ and $H$ is a subgroup of $G$ such that $[G:N]$ and $|H|$ are relatively primes, then $H\lt N$. I've already found $|H|||N|$ but I couldn't go ...
4
votes
1answer
34 views
Compute the splitting field
I have to compute the splitting field of $x^6-1 \in Q[x]$. I know that $x^6-1=(x+1)(x-1)(x^2-x+1)(x^2+x+1)$ but I don't know what to do after that.
Please help me.
1
vote
1answer
51 views
How to find the order of these groups?
I don't know why but I just cannot see how to find the orders of these groups:
$YXY^{-1}=X^2$
$YXY^{-1}=X^4$
$YXY^{-1}=X^3$
With the property that $X^5 = 1$ and $Y^4 =1$
How would I go about ...
3
votes
0answers
55 views
Finite locally groups
Let be $V\leq \operatorname{Aut}\left( G\right),\ N\vartriangleleft G$ and $V$-invariant.
Consider the semidirect product of $V$ with $G$. Let $C_{G}(V)=\{g\in G:g^{v}=g\}$ and $[G,V]=\langle ...
1
vote
1answer
32 views
Semi direct product-groups that are isomorphic
The questions asks me to find all groups up to isomorphism of the semi direct product $C_5 \rtimes C_4$
Now I've done the working out to get four groups (Note I've used $X$ as an element of $C_5$ and ...
0
votes
3answers
29 views
Prime ideal and nilpotent question
If $p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$, it follows that $r \in p$.
The only hint that my tutor gave me was to use induction. Can someone explain what he means by ...
0
votes
1answer
29 views
Prove that every element of $S_{n}$ can be written as a product $\sigma$ $= \tau_{1} . \tau_{2} … \tau_{r} $
Prove that every element of $S_{n}$ can be written as a product $\sigma$ $= \tau_{1} . \tau_{2} ... \tau_{r} $ of "flips"$ \\ (1,2)\ (2,3)\ ...\ (n-1,n) \ (n,1)$
of adjacent elements in the ...
5
votes
2answers
62 views
Can the associative property of a group be followed from its other properties?
Assume that we have a group together with inverse elements for all its group members, the closure property and also the identity. Can we follow from these properties that our group also has to fulfill ...
4
votes
0answers
43 views
Deciding whether or not a class of modules is “big enough”
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
1
vote
2answers
31 views
Show that r is a primitive root?
Show that if $r$ is a primitive root modulo the positive integer $m$, then $ {\overline r }$ is also a primitive root modulo m if $ {\overline r }$ is an inverse of $r$ modulo $m$.
My TA did not go ...
-1
votes
0answers
38 views
Is the group $\langle a, b \ | \ a^3, b^3, b^2a, ba^2, a^2b, ab^2\rangle$ normal in $\langle a, b\rangle$?
I am stuck on how to do the following problem. I am not sure if I am missing something, or whether this is actually a difficult problem.
How do I go about determining whether or not the group ...
3
votes
1answer
32 views
Galois group of irreducible quartic with real coefficients
Let $K$ be a subfield of the real numbers and $f\in K[x]$ be an irreducible quartic. If $f$ has exactly two real roots, show that the Galois group of $f$ is $S_{4}$ or $D_{4}$ (I'm using the ...
4
votes
0answers
29 views
Show that $ord(ab) | \frac{mn}{\gcd(m,n)}$ and $\frac{mn}{\gcd(m,n)^2}|ord(ab)$.
Suppose $G$ is an abelian group. Define $ord(a)=m$ and $ord(b)=n$ where $a,b \in G$. Show that $ord(ab) | \dfrac{mn}{\gcd(m,n)}$ and $\dfrac{mn}{\gcd(m,n)^2}|ord(ab)$.
Since ...
-3
votes
1answer
65 views
$\mathbb{Z}[\sqrt{-23}]$: A uniquely written set?
I suspect that
$\mathbb{Z}[\sqrt{-23}] \implies \forall~z=\sqrt{23b+a}~e^{i\arctan{\frac{23b}{a}}},~\text{where $z$ is uniquely written}~\forall~z\in \mathbb{Z}[\sqrt{-23}]$
5
votes
1answer
61 views
Non-central subgroup which is maximal among abelian normal subgroups is self-centralizing?
If $G$ is a supersolvabe group, and $A$ is a maximal among abelian normal subgroups of $G$, then the centralizer of $A$ in $G$ is $A$ itself (see link).
My question is about the importance of the ...
0
votes
0answers
54 views
A group of order 2 in a group of order 60.
If we assume that the 2, 3, and 5 Sylow subgroups of a group G of order 60 are not normal, how can I show that a group of order 2 in G cannot be normal?
I was trying to use contradiction. Assuming ...
3
votes
1answer
60 views
Irreducible polynomial over $\mathbb Z/7\mathbb Z$
Is the polynomial $t^6-3$ irreducible over $\mathbb{Z}/7\mathbb{Z}$? How would you prove that without doing all the computations? Is there a general method for this? Thank you.
1
vote
1answer
22 views
$f^{-1}(f(K))=KN$
I'm trying to prove If $f:G\to H$ is a homomorphism with Kernel $N$ and $K\lt G$, then $f^{-1}(f(K))=KN$.
Since $N$ is a normal subgroup of G, I know that $N\cap K$ is a normal subgroup of $K$ and ...
2
votes
2answers
59 views
Characterization of short exact sequences
The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah & Macdonald.
Let $A$ be a commutative ring with $1$. Let $$M'
...
1
vote
2answers
49 views
Example where $[E:\mathbb{Q}]<|\mathrm{Aut}_{\mathbb{Q}}E|$
Let $F$ be an algebraic closure of $\mathbb{Q}$ and let $E\subset F$ be a splitting field over $\mathbb{Q}$ of the set $\{x^{2}+a|a\in\mathbb{Q}\}$ so that $E$ is algebraic and Galois over ...
2
votes
1answer
68 views
Primes in $\mathbb{Z}[i]$
I need a bit of help with this problem
Let $x \in \mathbb{Z}[i]$ and suppose $x$ is prime, therefore $x$ is not a unit and cannot be written as a product of elements of smaller norm. Prove that ...
-2
votes
0answers
28 views
Prove that every finite group occurs as the Galois group of a field extension of the form…
$F(x_1,\ldots, x_n)/E$. Is $E$ supposed to be $F$ adjoined the elementary symmetric functions?
2
votes
3answers
54 views
$\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but not free over $\mathbb{Z}$.
I am struggling to explain the following statement to myself in a convincing way. $\mathbb{Z}/m\mathbb{Z}$ is free when considered as a module over itself, but is not free when considered as a ...
0
votes
1answer
54 views
Finding an isomorphism of fields
OK so basically the question asks me to find an explicit ring isomorphism
$ \alpha: x^2+x+2 \mapsto x^2+2x+2$
Where both are under the field of five elements ie (0,1,2,3,4). I don't know how to get ...
1
vote
1answer
26 views
bilinear form $F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$, find ortogonal subspaces, that satisfy…
Define $F$ as bilinear form $M_n(\mathbb{R}) \text{ x } M_n(\mathbb{R}) \rightarrow \mathbb{R}$
$F(A, B) = n \cdot \text{tr}(AB) - \text{tr}(A)\cdot\text{tr}(B)$
Prove, that $F$ is represented by ...
0
votes
5answers
85 views
Prove that $N$ is normal in $G$
I encountered this problem in Gallian Abstract Algebra (pg. 239, 8th edition)
Suppose that $G=H\times K$ and that $N$ is a normal subgroup of $H$. Prove that $N$ is normal in $G$.
Where $\times$ ...
0
votes
1answer
24 views
Smallest Galois extension
Let $F$ be a finite dimensional Galois extension of $K$ and let $E$ be an intermediate field. Show that there is a unique smallest field $L$ such that $E\subset L\subset F$ and $L$ is Galois over $K$. ...
3
votes
1answer
50 views
If $G/H$ is finitely generated, then so is $G$
I'm trying to prove that if $H$ is a normal subgroup of a group $G$ such that $H$ and $G/H$ are finitely generated, then G is finitely generated also. I'm trying to find a finite set $X$ such that $G$ ...
2
votes
1answer
29 views
Showing that a field extension is Galois
Let $F$ be an extension field of a field $K$. Let $E$ be an intermediate field such that $E$ is Galois over $K$, $F$ is Galois over $E$, and every $\sigma\in\mathrm{Aut}_{K}E$ is extendible to $F$. ...
