On Thompson's conjecture for alternating groups A p+3.

Area of Science:

• Group Theory
• Abstract Algebra
• Computational Group Theory

Background:

• The study builds upon previous work characterizing alternating groups (A_n) using properties of their prime graphs and conjugacy class element orders.
• The prime graph GK(G) connects primes dividing the group order |G| if an element of order pq exists.
• s(G) denotes the number of connected components in GK(G).

Purpose of the Study:

• To extend characterization theorems for finite groups.
• To investigate the relationship between the set of nonidentity orders of conjugacy classes (N(G)) and group isomorphism.
• Specifically, to determine if a finite group G with a trivial center is isomorphic to A_{p+3} given N(G) = N(A_{p+3}) and a composite p+2.

Main Methods:

• Utilizing concepts from group theory, including prime divisors of group orders, element orders, and conjugacy classes.
• Analyzing the structure of the prime graph GK(G) and its connected components (s(G)).
• Applying group isomorphism theorems and properties of alternating groups (A_n).

Main Results:

• The research proves that a finite group G with a trivial center is isomorphic to the alternating group A_{p+3} if its set of nonidentity conjugacy class element orders, N(G), is identical to that of A_{p+3}, provided that p+2 is a composite number.
• This extends existing characterization results for alternating groups.

Conclusions:

• The set of nonidentity orders of conjugacy classes, N(G), along with the trivial center condition and specific properties of p, can uniquely identify certain finite groups.
• The findings contribute to the understanding of group structure and classification through number-theoretic properties of group elements.