Lagrange's Theorem (Order of a group) Abstract Algebra

In summary, the conversation discusses various proofs and concepts related to abstract algebra, such as the order of a group, subgroups, cosets, and resources for learning more about the subject.
  • #1
anna010101
1
0
Can someones tells me how to prove these theorems.
1. Prove that if G is a group of order p^2 (p is a prime) and G is not cyclic, then a^p = e (identity element) for each a E(belongs to) G.

2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H.

3. Verify that S4 has at least one subgroup of order k for ech divisor of 24

4. If H is a subgroup of G and [G:H] = 2, the the right cosets of H in G are the same as the left cosets of H in G. Why?

5. Prove that if H is a subgroup of a finite grup G, the n the number of right cosets of H in G equals the number of left cosets of H in G.

Also, does anyone know a good website that has good information for abstract algebra. Thanks.
 
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  • #2
1. Prove ... a^p = e
Well, what else could that exponent be?

2. Prove that if H is a subgroup of G, [G:H]=2, a, b E G, a not E H and b not E H, then ab E H.
If you're only worried about whether or not something is in H... then why don't you work with G/H instead? (You know H is normal, right? If not, then see #4)

3. Verify that S4 has at least one subgroup of order k for ech divisor of 24
Just start writing down subgroups. I don't know what else to say.

4. If H is a subgroup of G and [G:H] = 2, the the right cosets of H in G are the same as the left cosets of H in G. Why?
You should know a very simple description of the left and right cosets of H. (Start by counting them -- you know how many there are)

5. Prove that if H is a subgroup of a finite grup G, the n the number of right cosets of H in G equals the number of left cosets of H in G.
Count. If you don't know what to count, then count everything you can imagine, in as many different ways as you can imagine.
 
  • #3
check my website.

or those of james milne, robert ash, ruslan sharipov, or just google your desired topic.
 

Related to Lagrange's Theorem (Order of a group) Abstract Algebra

1. What is Lagrange's Theorem in Abstract Algebra?

Lagrange's Theorem, also known as the Order of a Group Theorem, states that the order of a subgroup must divide the order of the parent group. In other words, the number of elements in a subgroup must be a factor of the number of elements in the entire group.

2. What is the significance of Lagrange's Theorem?

Lagrange's Theorem is a fundamental concept in Abstract Algebra and has many applications in various branches of mathematics, including number theory, geometry, and cryptography. It helps to determine the structure and properties of a group, making it an essential tool in the study of group theory.

3. How is Lagrange's Theorem applied in real-world problems?

Lagrange's Theorem is used in cryptography, specifically in the RSA algorithm. It is also used in coding theory to construct error-correcting codes. In addition, the theorem has applications in algebraic geometry, where it helps to classify algebraic curves.

4. What is the difference between the order of a group and the order of an element in a group?

The order of a group refers to the number of elements in the group, while the order of an element in a group refers to the smallest positive integer n such that the element, when raised to the power of n, equals the identity element of the group. In other words, the order of an element is the number of times it must be combined with itself to get the identity element.

5. Can Lagrange's Theorem be applied to infinite groups?

No, Lagrange's Theorem only applies to finite groups. This is because infinite groups do not have a well-defined order, making it impossible to determine the order of a subgroup in relation to the order of the entire group.

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