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Suppose that is a finite group with an element of order 5 and an element of order 7. Why must
The order of and the order must both divide the order of
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Suppose that is a finite group with an element of order 5 and an element of order 7. Why must
The order of and the order must both divide the order of
Suppose that is a finite group with 60 elements. What are the orders of possible subgroups of
The possible orders must divide 60.
Prove or disprove: Every subgroup of the integers has finite index.
This is true for every proper nontrivial subgroup.
Prove or disprove: Every subgroup of the integers has finite order.
False.
List the left and right cosets of the subgroups in each of the following.
in
in
in
in
in
in
in
in
(a) and (c) and
Describe the left cosets of in What is the index of in
Verify Euler's Theorem for and
Use Fermat's Little Theorem to show that if is prime, there is no solution to the equation
Show that the integers have infinite index in the additive group of rational numbers.
Show that the additive group of real numbers has infinite index in the additive group of the complex numbers.
Let be a subgroup of a group and suppose that Prove that the following conditions are equivalent.
If for all and show that right cosets are identical to left cosets. That is, show that for all
Let Show that and thus
What fails in the proof of Theorem 6.8 if is defined by
Suppose that Show that the order of divides
Show that any two permutations have the same cycle structure if and only if there exists a permutation such that If for some then and are .
If prove that the number of elements of order 2 is odd. Use this result to show that must contain a subgroup of order 2.
Suppose that If and are not in show that
If prove that
Let and be subgroups of a group Prove that is a coset of in
Show that
Let and be subgroups of a group Define a relation on by if there exists an and a such that Show that this relation is an equivalence relation. The corresponding equivalence classes are called . Compute the double cosets of in
Let be a cyclic group of order Show that there are exactly generators for
Show that
for all positive integers