Proving the Inclusion of <a> in H: A Permutation Group Proof

In summary, the problem is to prove that if a is an element of a group G and H is a subgroup of G, then a is in H if and only if the set <a>, generated by a, is a subset of H. The proof involves showing that if a is in H, then <a> is a subset of H, and if <a> is a subset of H, then a is in H. This can be easily demonstrated by considering the properties of groups and the definition of <a>.
  • #1
iamalexalright
164
0

Homework Statement


Suppose G is a group, H < G (H is a subgroup of G), and a is in G.

Prove that a is in H iff <a> is a subset of H.


Homework Equations


<a> is the set generated by a (a,aa,aa^-1,etc)


The Attempt at a Solution


For some reason this seems too easy:

1. Suppose a is in H.
Since H is a group, a^-1 is in H.
Since H is a group aa, is in H (as is aa^-1, etc.)
Thus <a> is a subset of H.

2. Suppose <a> is a subset of H.
Obviously a is in H.

And this completes the proof... or am I missing something?
 
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  • #2
shameful bump
 

1. What is a permutation group?

A permutation group is a mathematical concept that describes a set of objects and the different ways those objects can be rearranged or permuted. This group is closed under composition, meaning that the result of combining two permutations from the group is another permutation in the group.

2. How are permutation groups used in proofs?

Permutation groups are often used in proofs to show that certain mathematical statements or equations are true. By rearranging the elements in a specific way, a permutation group can demonstrate the validity of a given statement.

3. What is the order of a permutation group?

The order of a permutation group is the number of elements or permutations in the group. It is denoted by |G|, where G is the group. This number can help determine the complexity of the group and its elements.

4. Can any set of objects be a permutation group?

No, not all sets of objects can form a permutation group. For a set to be considered a permutation group, it must meet certain criteria, such as closure under composition, having an identity element, and every element having an inverse.

5. Are all permutation groups the same?

No, not all permutation groups are the same. Permutation groups can vary in size, complexity, and structure, depending on the specific set of objects and the rules for rearranging them. Some permutation groups may also have special properties, such as being abelian or non-abelian.

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