FINAL SOLUTIONS

# Algebra

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Math 150a: Modern Algebra Solutions to the Final 1. If G is a group with a subset A , then conditions for A to be a subgroup are: (1) A is closed under multiplication, (2) A contains the identity 1, and (3) A is closed under taking inverses. Suppose that G is a finite group and that A is a non-empty subset. Prove that in this case, if A is closed under multiplication, the other two conditions follow automatically. Solution: From Huilin Chen: She did not quite spell out explicitly that g - 1 arises in A as g n - 1 , but from the displayed formula it gets the benefit of the doubt.

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2. Is the matrix A = parenleftbigg 1 - 2 3 4 parenrightbigg an element of the group GL ( 2 , Z / 5 ) ? How about in GL ( 2 , Z / 7 ) ? Find its inverse (using the standard names of the elements in Z / n ) if the answer is yes in either case. Solution: From Sonny Lau: The calculation got chopped a little bit, but the missing part is straightforward.
3. In the group S 5 , let A be the subgroup of permutations that fix the number 1, and let B be the subgroup of permutations that fix the number 5.

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