15.5: A Project

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The main objective of finite group theory is to classify all possible finite groups up to isomorphism. This problem is very difficult even if we try to classify the groups of order less than or equal to \(60\text{.}\) However, we can break the problem down into several intermediate problems. This is a challenging project that requires a working knowledge of the group theory you have learned up to this point. Even if you do not complete it, it will teach you a great deal about finite groups. You can use Table \(15.21\) as a guide.

\(Table \text { } 15.21\). Numbers of distinct groups \(G\text{,}\) \(|G| \leq 60\)

Order	Number	Order	Number	Order	Number	Order	Number
\(1\)	?	\(16\)	\(14\)	\(31\)	\(1\)	\(46\)	\(2\)
\(2\)	?	\(17\)	\(1\)	\(32\)	\(51\)	\(47\)	\(1\)
\(3\)	?	\(18\)	?	\(33\)	\(1\)	\(48\)	\(52\)
\(4\)	?	\(19\)	?	\(34\)	?	\(49\)	?
\(5\)	?	\(20\)	\(5\)	\(35\)	\(1\)	\(50\)	\(5\)
\(6\)	?	\(21\)	?	\(36\)	\(14\)	\(51\)	?
\(7\)	?	\(22\)	\(2\)	\(37\)	\(1\)	\(52\)	?
\(8\)	?	\(23\)	\(1\)	\(38\)	?	\(53\)	?
\(9\)	?	\(24\)	?	\(39\)	\(2\)	\(54\)	\(15\)
\(10\)	?	\(25\)	\(2\)	\(40\)	\(14\)	\(55\)	\(2\)
\(11\)	?	\(26\)	\(2\)	\(41\)	\(1\)	\(56\)	?
\(12\)	\(5\)	\(27\)	\(5\)	\(42\)	?	\(57\)	\(2\)
\(13\)	?	\(28\)	?	\(43\)	\(1\)	\(58\)	?
\(14\)	?	\(29\)	\(1\)	\(44\)	\(4\)	\(59\)	\(1 \)
\(15\)	\(1\)	\(30\)	\(4\)	\(45\)	?	\(60\)	\(13\)

1

Find all simple groups \(G\) ( \(|G| \leq 60\)). Do not use the Odd Order Theorem unless you are prepared to prove it.

2

Find the number of distinct groups \(G\text{,}\) where the order of \(G\) is \(n\) for \(n = 1, \ldots, 60\text{.}\)

3

Find the actual groups (up to isomorphism) for each \(n\text{.}\)