Group Theory for Symmetric Groups
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1) Recall that So denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6 ... see attached for full questions
1) Recall that denotes the symmetric group of degree 6, the group of permutations f the numbers 1 to 6. let .
Thus is a bijection, mapping 1 to 3, 2 to 5, etc. let
• Compute and and write them also b array form. (don't forget that, as we write maps in the left , means do and then , so and so on)
• Compute the order of .
• Elements of can be written in an alternative form, called cycle notation. Starting with 1, we see that , back to the start. So the first cycle for is (1 3 4). As this has 3 numbers, it is called 3-cycle. Next, we take a number not yet mentioned, e.g, 2. we see that . So the next cycle for is (2 5). Finally we get a 1-cycle, (6). We write . Write in cycle notation.
Please see the attachment.
1. First, we count . We note
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