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.
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34 views
Group presentations and subgroups
How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$?
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1answer
30 views
Subgroup complement for normal subgroup of $G$ with trivial center and $\mathrm{Aut}(N)=\mathrm{Inn}(N)$
Let $G$ be a finite group and $N \subseteq G$ be a normal subgroup. If $Z(N)$ is trivial and $\operatorname{Aut} N=\operatorname{Inn} N$, then show $N$ has a complement $H$ and $H$ is normal in $G$ ...
4
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0answers
41 views
Algebraic structure of a set of Egyptian fractions of a positive rational?
It is said that every positive rational number can be represented by infinitely many Egyptian fractions (defined as the sum of distinct unit fractions).
I am struggling to understand in a formal way, ...
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votes
4answers
53 views
A presentation of a group of order 12
Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$.
I tried to let $d=ab\Rightarrow G=\langle d,c\mid ...
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2answers
34 views
How to make existence proof for Abelian Group condition 3( unit element e), when ($\mathbb{N}. \cdot)$?
How to make existence proof for Abelian Group condition 3( unit element e), when ($\mathbb{N}. \cdot )$, where $\cdot$ is natural multiplication?
Is it by example? Is it done by constructive proof ...
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1answer
20 views
upper central series, taking intersection with subgroup ($\zeta_i G \cap H \leq \zeta_i H$)
Let $\zeta_i G$ be the upper central series of $G$. Show that for 'any' subgroup $H$, we have
$$\zeta_i G \cap H \leq \zeta_i H$$
where $\zeta_i H$ is the upper central series of $H$.
I tried ...
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2answers
56 views
Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$
$p$ is a prime number, $k$ is an positive integer, and $f\in\Bbb Z[x]$.
Prove: $f(x)^{p^k}\equiv f\left(x^{p^k}\right)\bmod p$
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1answer
32 views
question on group representations
Here is a problem I faced in algebra. $\rho: A_4 \rightarrow End_{\mathbb C}\mathbb C^{10}$ is a representation of $A_4$. Then show that there is a vector $v \in \mathbb C^{10}$ such that $v$ is an ...
1
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1answer
38 views
To What Extent Does the Cartesian Product for Algebraic Structures Generalize?
I admit this question is quite general.
If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
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2answers
45 views
A prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ so that $7\in I$.
Find a prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ such that $7\in I$.
I claimed that $I= 7\mathbb Z[\sqrt{-5}]$, and tried to prove that if $x,y\in \mathbb Z[\sqrt{-5}]$ and so that $xy\in I$ then ...
4
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1answer
55 views
Let $F= \Bbb{Q}(\zeta_9)$ with $\zeta_9 = e^{\frac{2i\pi}{9}}$. Find the Galois group and the intermediate fields.
Let $F= \Bbb{Q}(\zeta_9)$ with $\zeta_9 = e^{\frac{2i\pi}{9}}$.
a) What is the Galois group of $F$ over $\Bbb{Q}$?
b) Find all intermediate fields between $\Bbb{Q}$ and $F$. (Write each in ...
0
votes
1answer
74 views
Check existence of an isomorphism in four examples
Check if there exists an isomorphism for :
$$ i) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, +_{15}), \ f(1) = 2 $$
$$ ii) \ G = (\mathbb{Z}_{15}, +_{15}), \ H = (\mathbb{Z}_{15}, ...
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2answers
33 views
Frieze groups, wallpaper groups
Can someone suggest a source that proves the classifications of the 7 frieze groups and 17 wallpaper groups in an elegant way?
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1answer
40 views
A submodule of a free module is torsion-free?
I am studying for a comprehensive exam and looking at a large bank of problems. One problem has six statements about module and asks for a a proof or a counter-example of the statements. I am able ...
3
votes
2answers
54 views
Quotient field of a certain quotient ring
Let $A$ be a commutative integral domain and $\mathfrak p$ a prime ideal of $A$. Let $A_{\mathfrak p}$ be the localization of $A$ at $\mathfrak p$ and $\mathfrak{m}_{\mathfrak{p}}=\mathfrak ...
