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If H, K are two subgroups of a group G, then H K is a subgroup of G, iff
- a)HK = 1
- b)HK = KH
- c)HK = H-1 K-1
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
If H, K are two subgroups of a group G, then H K is a subgroup of G, i...
Explanation:
To prove that option 'B' is the correct answer, we need to show that if HK is a subgroup of G, then HK = KH.
Proof:
Let's assume that HK is a subgroup of G.
1. HK is a subset of KH:
For any h in H and k in K, we have hk in KH. Therefore, HK is a subset of KH.
2. KH is a subset of HK:
For any k in K and h in H, we have kh in HK. Therefore, KH is a subset of HK.
3. HK = KH:
From (1) and (2), we can conclude that HK is equal to KH. Hence, option 'B' is the correct answer.
Alternative Proof:
We can also prove this result using the fact that if H and K are subgroups of G, then HK is a subgroup of G if and only if HK = KH.
Proof:
1. HK is a subgroup of G:
Let's assume that HK is a subgroup of G. In order to prove that HK = KH, we need to show that HK is a subset of KH and KH is a subset of HK.
2. HK is a subset of KH:
For any h in H and k in K, we have hk in KH. Therefore, HK is a subset of KH.
3. KH is a subset of HK:
For any k in K and h in H, we have kh in HK. Therefore, KH is a subset of HK.
4. HK = KH:
From (2) and (3), we can conclude that HK is equal to KH. Hence, option 'B' is the correct answer.
Therefore, the correct answer is option 'B' (HK = KH).
To prove that option 'B' is the correct answer, we need to show that if HK is a subgroup of G, then HK = KH.
Proof:
Let's assume that HK is a subgroup of G.
1. HK is a subset of KH:
For any h in H and k in K, we have hk in KH. Therefore, HK is a subset of KH.
2. KH is a subset of HK:
For any k in K and h in H, we have kh in HK. Therefore, KH is a subset of HK.
3. HK = KH:
From (1) and (2), we can conclude that HK is equal to KH. Hence, option 'B' is the correct answer.
Alternative Proof:
We can also prove this result using the fact that if H and K are subgroups of G, then HK is a subgroup of G if and only if HK = KH.
Proof:
1. HK is a subgroup of G:
Let's assume that HK is a subgroup of G. In order to prove that HK = KH, we need to show that HK is a subset of KH and KH is a subset of HK.
2. HK is a subset of KH:
For any h in H and k in K, we have hk in KH. Therefore, HK is a subset of KH.
3. KH is a subset of HK:
For any k in K and h in H, we have kh in HK. Therefore, KH is a subset of HK.
4. HK = KH:
From (2) and (3), we can conclude that HK is equal to KH. Hence, option 'B' is the correct answer.
Therefore, the correct answer is option 'B' (HK = KH).
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If H, K are two subgroups of a group G, then H K is a subgroup of G, iffa)HK = 1b)HK = KHc)HK = H-1K-1d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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