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### A Classification of Groups of Small Order upto Isomorphism

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A Classification of Groups of Small Order upto Isomorphism

Ezenwobodo Somkene Samuel

Ezenwobodo Somkene Samuel "A Classification of Groups of Small Order upto Isomorphism" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4, June 2020, pp.627-631, URL: https://www.ijtsrd.com/papers/ijtsrd31139.pdf

Here we classified groups of order less than or equal to 15. We proved that there is only one group of order prime up to isomorphism, and that all groups of order prime (P) are abelian groups. This covers groups of order 2,3,5,7,11,13….Again we were able to prove that there are up to isomorphism only two groups of order 2p, where p is prime and p=3, and this is Z_2p ? Z_2 x Z_p. (Where Z represents cyclic group), and D_p (the dihedral group of the p-gon). This covers groups of order 6, 10, 14….. And we proved that up to isomorphism there are only two groups of order P2. And these are Z_(p^2 ) and Z_p x Z_p. This covers groups of order 4, 9…..Groups of order P3 was also dealt with, and we proved that there are up to isomorphism five groups of order P3. Which areZ_(p^3 ), Z_(p^2 ) x Z_p, Z_p x Z_p x Z_p, D_(p^3 ) and Q_(p^3 ). This covers for groups of order 8… Sylow’s theorem was used to classify groups of order pq, where p and q are two distinct primes. And there is only one group of such order up to isomorphism, which is Z_pq ? Z_p x Z_q. This covers groups of order 15… Sylow’s theorem was also used to classify groups of order p^2 q and there are only two Abelian groups of such order which are Zp2q and Z_p x Z_p x Z_q. This covers order 12. Finally groups of order one are the trivial groups. And all groups of order 1 are abelian because the trivial subgroup of any group is a normal subgroup of that group.

Abelian, cyclic, isomorphism, order, prime.

IJTSRD31139
Volume-4 | Issue-4, June 2020
627-631
IJTSRD | www.ijtsrd.com | E-ISSN 2456-6470