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Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

Abstract algebra is the study of algebraic objects, i.e. sets endowed with one or more operations on the elements of those sets. In particular, the study of abstract algebra considers the algebraic structures and properties of which such operations induce. It can be considered as the generalization of the study of the algebraic structure of the integers and real numbers (arithmetic), or the study of matrices and vector spaces (linear algebra).

Some algebraic objects are monoids, groups, rings, fields, vector spaces, modules, algebras, and categories, among many other less prominent objects.

Examples

  1. The set of non-negative integers $\mathbb{N} = \{0,1,2,3,\dotsc\}$ is a monoid under the operation $+$.

  2. The integers $\mathbb{Z} = \{\dotsc,-1,0,1,\dotsc\}$ under the binary operation of $+$ form a group.

  3. Furthermore, $\mathbb{Z}$ has the structure of a ring when you consider it as being equipped with both addition and multiplication.

  4. The real numbers $\mathbb{R}$ with their usual addition and multiplication form a field.

  5. The set of $n\times n$ matrices with entries in $\mathbb{R}$ with matrix addition and multiplication form a ring.

  6. The set of $1\times n$ vectors over the real numbers, with vector addition, and multiplication by elements of the $n\times n$ real matrices on the right are an example of a module for the ring of matrices.

In addition to studying the objects themselves, abstract algebra considers homomorphisms between the objects and various constructions and tools, which are useful for studying the objects.

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I study abstract algebra mainly because I'm planning to study relativity Please give me a guideline for learning this subject

I'm not duplicating how to study abstract algebra. I recently downloaded Herstein's Abstract algebra and there are many contents in the book and I actually want to know if I need to read through all of the chapters if I learn it just for relativity.…
pxc3110
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Is there a size of group for which no group is simple but it's possible that no Sylow subgroup is normal?

It seems like proving that groups of a certain size are never simple is usually done with Sylow theorems, showing that a Sylow subgroup of a particular size must be normal. But is there an example group size $n$ where no groups of size $n$ are…
user2566092
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groups, subgroups, cosets

I'm still a little confused about cosets. Maybe someone could look my proof over, please? If $G$ is a group and $H$ is a subgroup, $a, b \in G$, need to prove $Ha = H$, iff $a \in H$. (->) Let $Ha = H$. Suppose $s \in Ha$. Then $s = ha$ for some…
Bob
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If two cycles are disjoints, then they commute

I'm trying to prove that two $\tau$-cycles commute provided that these cycles are disjoints. Hungerford in his book says the following remark about this fact: Intuitively clearly this is true, but how can we prove this formally? Thanks in advance
user42912
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Proving that $|A_n| = n!/2$

I can't figure out why we need $\phi$ to be bijective in order to conclude $|A_n| = |S_n| - |A_n|$, could someone please explain how this works?
Warz
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irreducibility of $t^5-t+1.$

In p.20 Stewart and Tall's algebraic number theory, it says that the polynomial $t^5-t+1$ is irreducible: consider $mod 5$, there is no linear factor since none of $0,1,2,3,4$ when substituted fir $t$, so the only possible way to factorize is…
taylor
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Are these two groups isomorphic?

I'm looking at Section 11, problem 18 in Fraleigh. Here's the question: Is $Z_8 \times Z_{10} \times Z_{24}$ isomorphic to $Z_4 \times Z_{12} \times Z_{40}$? I can do the problem once I figure classify each according to the fundamental theorem of…
anon_swe
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Prove for each $a ∈ G, aHa^{-1}$ is a subgroup of G.

Question: Let $H$ be a subgroup of $G$. For any $a \in G$, let $aHa^{-1} = \{axa^{-1} : x \in H\}$; $aHa^{-1}$ is called a conjugate of $H$. Prove: For each $a \in G$, $aHa^{-1}$ is a subgroup of $G$. I know in order to prove something is a subgroup…
Bradley
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Trying to determine if $G = \mathbb{R}$ then $H = \{\log a \mid a \in \mathbb{Q}, a > 0\}$ is a subgroup.

I am trying to prove that if $G = \mathbb{R}$ then $H = \{\log a \mid a \in \mathbb{Q}, a > 0\}$ is a subgroup. The identity of $G$ is $0$ and $0 \in H$. If $a,b \in \mathbb{Q}$ and $a>0$ and $b>0$ then $\log ab = \log a + \log b$. If $x \in H$…
spitfiredd
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General Algorithm for Computing Factor Groups

I'm a bit confused on computing factor groups. Fraleigh defines it as classifying a factor group according to the fundamental theorem (saying what the factor group is isomorphic to). For instance, in Example 15.7 he has: Compute the factor group…
anon_swe
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Find the centralizer of (1 2) in S5.(Conjugation)

Find the centralizer of (1 2) in S5. (Conjugation in the symmetric group) I have no idea how to solve this kind of problem. Can you show me the method? Thanks!
Curran
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what is the difference between product and sum of groups?

The direct product of two groups equals their direct sum. What is the difference in the case of infinite product and sum?
user3474383
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Find $a$ and $b$ with whom this expression $x\bullet y=(a+x)(b+y)$ is associative

I need to find a and b with whom this expression is associative: $$x\bullet y=(a+x)(b+y)$$ Also known that $$x,y\in Z$$ So firstly I write: $$(x\bullet y)\bullet z=x\bullet (y\bullet z)$$ Then I express and express them and after small algebra I…
Templar
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Number of order 5 elements in $(\mathbb{Z}/31\mathbb{Z})^{\times}$

I know there are $4$ elements of order $5$ and the elements are $\bar{2}$,$\bar{4}$,$\bar{8}$,$\bar{16}$. I was only able to show it by using brute force. Eg. I found that the element $\bar{2}$ has order $5$ by calculating $2^5=32 \equiv 1 \…
user114653
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Operation on a Group that isn't stated

If we are given $G = \mathbb{R}$ and the operation isn't explicitly given, is it assumed to be the $+$ operator?
spitfiredd
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