Is HK a Subgroup of S_5? - Homework Statement & Equations

e(ho0n3 · Nov 11, 2008

Homework Statement

Let H, K be subgroups of S_5, where H is generated by (1 2 3) and K is generated by (1 2 3 4 5). Is HK a subgroup of S_5?

Homework Equations

HK is a subgroup iff HK = KH.

The Attempt at a Solution

Is there an easy way of answering this question without computing HK and KH?

e(ho0n3 · Nov 11, 2008

Maybe the simplest thing to try is to show that HK is a subgroup directly from the definition of subgroup.

HK is obviously non-empty. Let x = (1 2 3), let y = (1 2 3 4 5) and let a = xⁱy^j and b = x^ky^m be elements of HK.

Then ab^-1 = xⁱy^j-mx^-k. Let n be such that k + n = i. Then xⁱy^j-mx^-k = x^k(xⁿy^j-m)x^-k = cd where c is in H and d is in K, since cojugating preserves cycle structure. Right? Thus, ab^-1 is in HK and HK is a subgroup.

Dick · Nov 11, 2008

I don't believe that proof for a minute. The product of the two groups contains all kinds of cycle structure. Like HK contains (3,4,5). That's a 3-cycle and it's not in H. And I'll tell you another thing. (3,4,5) is not in KH. I haven't figured out any special tricks to know that yet. I did it the hard way.

e(ho0n3 · Nov 11, 2008

Coming from you, that's not a good sign. Can you tell me the flaw in my proof?

Dick · Nov 11, 2008

e(ho0n3 said:

Coming from you, that's not a good sign. Can you tell me the flaw in my proof?

I can't really tell you because I don't see how it proves anything. You seem to be thinking everything in HK has the cycle structure of a disjoint three cycle and a five cycle. And the three cycle must belong to H and the five cycle must belong to K. Is that what you are thinking? I can only guess. It's not true. (1,2,3)*(1,3,5,2,4)=(2,4)*(3,5). So?

e(ho0n3 · Nov 11, 2008

That's exactly what I was thinking. I realize now that it is wrong. Oh well.

Dick · Nov 11, 2008

e(ho0n3 said:

That's exactly what I was thinking. I realize now that it is wrong. Oh well.

That doesn't mean there isn't some trick you can use to show HK is not equal to KH without evaluating all 15 products in both sets (which really isn't that bad, only 8 of them are nontrivial in each set). It just means I haven't thought of one.

Is HK a Subgroup of S_5? - Homework Statement & Equations

Homework Statement

Homework Equations

The Attempt at a Solution

Related to Is HK a Subgroup of S_5? - Homework Statement & Equations

1. Is HK a subgroup of S₅?

Yes, HK is a subgroup of S₅ if and only if it satisfies the two conditions of closure and inverses. If every element in HK is also in S₅ and every element in HK has an inverse in HK, then HK is a subgroup of S₅.

2. What is the significance of S₅ in this question?

S₅ is the symmetric group of order 5, which means it consists of all possible permutations of 5 objects. It is relevant in this question because HK is being compared to it as a potential subgroup.

3. What is the definition of a subgroup?

4. How can we determine if HK is a subgroup of S₅?

To determine if HK is a subgroup of S₅, we can check if it satisfies the two conditions of closure and inverses. This can be done by checking if every element in HK is also in S₅ and if every element in HK has an inverse in HK.

5. What are the possible outcomes of this question?

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Is HK a Subgroup of S_5? - Homework Statement & Equations

Homework Statement

Homework Equations

The Attempt at a Solution

Related to Is HK a Subgroup of S_5? - Homework Statement & Equations

1. Is HK a subgroup of S5?

Yes, HK is a subgroup of S5 if and only if it satisfies the two conditions of closure and inverses. If every element in HK is also in S5 and every element in HK has an inverse in HK, then HK is a subgroup of S5.

2. What is the significance of S5 in this question?

S5 is the symmetric group of order 5, which means it consists of all possible permutations of 5 objects. It is relevant in this question because HK is being compared to it as a potential subgroup.

3. What is the definition of a subgroup?

4. How can we determine if HK is a subgroup of S5?

To determine if HK is a subgroup of S5, we can check if it satisfies the two conditions of closure and inverses. This can be done by checking if every element in HK is also in S5 and if every element in HK has an inverse in HK.

5. What are the possible outcomes of this question?

Similar threads

Hot Threads

Recent Insights

1. Is HK a subgroup of S₅?

Yes, HK is a subgroup of S₅ if and only if it satisfies the two conditions of closure and inverses. If every element in HK is also in S₅ and every element in HK has an inverse in HK, then HK is a subgroup of S₅.

2. What is the significance of S₅ in this question?

S₅ is the symmetric group of order 5, which means it consists of all possible permutations of 5 objects. It is relevant in this question because HK is being compared to it as a potential subgroup.

4. How can we determine if HK is a subgroup of S₅?

To determine if HK is a subgroup of S₅, we can check if it satisfies the two conditions of closure and inverses. This can be done by checking if every element in HK is also in S₅ and if every element in HK has an inverse in HK.