Abstract Algebra; Group Theory Question

Thread starter tropian1
Start date Mar 11, 2015
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Abstract Abstract algebra Algebra Group Group theory Theory

In summary, Abstract Algebra is a branch of mathematics that focuses on the study of algebraic structures, such as groups, rings, fields, and vector spaces. Group Theory, which is a subset of Abstract Algebra, specifically studies the mathematical concept of a group, which is a set of elements with a binary operation that satisfies certain properties. Group Theory has many real-life applications, including in physics, chemistry, computer science, and cryptography. The main properties of a group are closure, associativity, identity, and invertibility. In comparison, a ring has two binary operations, addition and multiplication, and only certain elements have an inverse. A ring must also satisfy the distributive law, whereas a group does not have this requirement.

Mar 11, 2015

tropian1

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Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f.

I'm having trouble defining a function to prove this. Could anyone give me a start on this?

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Mar 11, 2015

micromass

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Start by proving the following:
1) ##N## and ##K## are normal
2) ##NK = G##
3) ##N\cap K = \{e\}##

1. What is Abstract Algebra?

Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, fields, and vector spaces. It focuses on the abstract properties that these structures possess rather than specific numbers or objects.

2. What is Group Theory?

Group Theory is a branch of Abstract Algebra that studies the mathematical concept of a group. A group is a set of elements together with a binary operation that satisfies certain axioms, such as closure, associativity, identity, and invertibility.

3. How is Group Theory used in real life?

Group Theory has numerous applications in different fields, including physics, chemistry, computer science, and cryptography. It is used to study the symmetry of physical systems, analyze molecular structures, design error-correcting codes, and secure communication protocols.

4. What are the main properties of a group?

The main properties of a group are closure, associativity, identity, and invertibility. Closure means that the result of combining any two elements in the group is also in the group. Associativity means that the order of operations does not matter. Identity means that there is an element in the group that acts as a neutral element. Invertibility means that every element in the group has an inverse that when combined, gives the identity.

5. What is the difference between a group and a ring?

A group is a mathematical structure that consists of a set of elements and a binary operation, while a ring is a structure that has two binary operations, addition and multiplication. In a group, every element has an inverse, but in a ring, only the elements that have a multiplicative inverse have an additive inverse. Additionally, a ring must satisfy the distributive law, while a group does not have this requirement.