Classification of simple groups upto order 200

Introduction Basic Terminology

Group: In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. Mathematically A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms

Closure property

For all a, b in G, the result of the operation, a • b, is also in G.

Associative property

Identity element

There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. Such an element is unique and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted a -1 (or-a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element. The result of an operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a.

Abelian Group: A group G is said to be abelian if ab = ba for all a,b G. Subgroup: Let G be a group, and let H be a subset of G. Then H is called a subgroup of G if H is itself a group, under the operation induced by G.

Normal Subgroup. In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G; i.e., the sets of left and right cosets coincide.

Simple group:

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated. p-subgroup: When is a prime number, then a p-group is a group, all of whose elements have order some power of or equivalent definition is that whose is order is some power of p Sylow p-subgroup: If is the highest power of a prime dividing the order of a finite group G, then any subgroup of of order is called a Sylow -subgroup of G.

Theorem 1.1 A group of prime order is always simple.

Proof: As we know that a prime number has namely two divisors that are only 1 and prime number itself. So as by Lagrange's theorem the order of subgroup must divide the order of group so if at all a group of prime order has a subgroup it must be of order 1 or p {where p = o(G)}. In fact these are the only subgroups which a group of prime order has and these are also the normal subgroups as we know that every group has two normal subgroups. So a group of prime order does not have any proper normal subgroup so it is not simple. Thus whenever we are talking of checking whether the group is simple or not we must consider only groups of Composite order as aa the group of prime order are always simple.

Theorem 1.2 An abelian group G of composite order can not be simple Proof: As we know that every subgroup of an abelian group G is a normal subgroup Also the converse of lagrange's theorem holds for finite abelian groups. Therefore if we have a abelian group of composite order say n it must have a proper divisor say k so by converse of lagrange,s theorem it must have subgroup of order k. Also since it is abelian therefore that subgroup of order k is normal also. Thus we have a proper normal subgroup of G. Thus G is not simple.

Theorem 2.1 Sylow's First theorem: If G is a finite group of order p n q (n≥1), where is a prime number and q is any positive integer such that (p,q) = 1. Then for each i (1≤i ≤ n), G has at least one subgroup of order p n i.e a sylow p-subgroup. Thus Sylow's first theorem shows the existence of sylow p-subgroups. Corollary 2.3.1 :Since we can always write the order of group in the form p n q where (p,q)=1 then by above theorem the number of sylow p-subgroups is 1+kp where 1+kp | o(G) i.e1+kp | p n q and (p,q) = 1 i.e 1+kp | q. Thus to conclude by using sylow third theorem the number of sylow p-subgroups is 1+kp where 1+kp | q.

Theorem 2. 4 If there is only one Sylow p-subgroup of a group then it is always normal subgroup of that group Proof: Let o (G) = p n q where (p,q)=1 Let P be the only Sylow p-subgroup of G so that o (G)= p n . Now g -1 Pg is also a subgroup of G g  G and o (g -1 Pg) = o (P) thus g -1 Pg is also a Sylow p-subgroup but as we are already given that there is only one Sylow p-subgroup so both must coincide. Thus g

Theorem 2.7

If G is a group such that o (G) =pq where pq p,q are primes and p ∤ q-1. Then G is cyclic Proof: By Sylow's First Theorem G has atleast on Sylow p-subgroup and one Sylow q-subgroup. By Sylow's Second Theorem, the number of Sylow p-subgroup is 1+kp where 1+kp|q Thus 1+kp = 1 or q If 1+kp = 1 then sylow p-subgroup is unique hence normal in G hence G has a proper Normal subgroup thus G is not simple then we are through Again by Sylow's Second Theorem, the number of Sylow q-subgroup is 1+kq where 1+k'q|p Thus 1+k'q = 1 or p If 1+k'q = 1 then sylow q-subgroup is unique hence normal in G hence G has a proper Normal subgroup thus G is not simple again we are through If 1+k'q = p then k'q = p -1 which means that q |p -1 which is not true. Now o (H) = p, o (K) = q, and H ∩ K is a subgroup of both H and K.

References

1. Singh Surjeet, Qazi Zameeruddin, Modern Algebra. Vikas Publishing House, New Delhi (8 th Edition), 2006.
2. Artin Algebra M. Prentice Hall of India, New Delhi, 1994.