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Introduction to Groups
- Determine if a given set, with respect to a given operation, forms a group.
- Determine properties of permutations.
- Decide if a given subset is a subgroup.
- Determine the group of symmetries of a given figure.
Equivalence, Congruence and Divisibility
- Verify properties of an equivalence relation.
- Find representatives of equivalence classes.
- Identify congruence classes modulo n .
- Perform the division algorithm.
- Apply group properties to Zn .
- Use the Euclidean Algorithm to prove statements involving greatest common divisors.
- Compute the greatest common divisor and least common multiple of pairs of integers.
Groups: Lagrange’s Theorem and Isomorphism
- Determine the cosets of a given group.
- Determine if given groups are isomorphic.
- Use Cayley’s theorem to find permutations associated with an element of a group.
- Determine subgroups for a given group and its elements’ order.
- Identify generators and direct products of groups.
- Find all subgroups of a given group using Lagrange’s theorem and its corollaries.
- Determine an action of a finite group on a set.
- Apply Sylow’s theorem to calculate Sylow subgroups.
- Use Burnside’s Counting theorem to determine the number of orbits for a group acting on a set.
- Determine if a given mapping is a homomorphism of groups.
- Find a kernel of a homomorphism.
- Identify normal subgroups for a given group and quotient groups.
- Verify relationships between different groups using the Fundamental Homomorphism theorem.
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