Group axioms, subgroups, cyclic groups, permutation groups, Lagrange theorem, homomorphisms, normal subgroups and quotient groups.
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- What are the four axioms that define a group?
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Closure, associativity, identity element, and inverse element.- What does the closure axiom state?
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If a and b are in the group G, then a * b is also in G.- What does the associativity axiom state?
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For all a, b, c in G, (a * b) * c = a * (b * c).- What is the identity element of a group?
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An element e in G such that a * e = e * a = a for all a in G.- What is the inverse element of a group element a?
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An element a^-1 in G such that a * a^-1 = a^-1 * a = e.- What is an abelian group?
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A group where the operation is commutative: a * b = b * a for all a, b in G.- What is a subgroup?
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A non-empty subset H of a group G that is itself a group under the same operation.- What is the one-step subgroup test?
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A non-empty subset H of G is a subgroup if and only if a * b^-1 is in H for all a, b in H.- What is the two-step subgroup test?
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A non-empty subset H of G is a subgroup if and only if H is closed under the operation and contains inverses of all its elements.- What are the trivial subgroups?
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The identity element {e} and the group itself G, which are always subgroups of any group.- What is a cyclic group?
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A group generated by a single element a, meaning every element can be written as a^n for some integer n.- What is a generator of a cyclic group?
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An element a in a cyclic group G such that every element of G is a power of a.- What is the order of an element a in a group?
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The smallest positive integer n such that a^n = e, or infinity if no such n exists.- What is a cyclic group of order n?
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A cyclic group with n elements, generated by an element of order n.- How do finite and infinite cyclic groups differ?
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Finite cyclic groups have all non-identity elements of finite order, while infinite cyclic groups have only the identity with finite order.
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