Question

Consider Z15, the group under addition modulo Z15. Let H1 =[0,5,10], H2 = [0,4,8,12].Are H1andH2 subgroup of Z15 under Z+15?

Answer 1

Answer:

H₁ is a subgroup of Z₁₅ whereas H₂ is not a subgroup of Z₁₅

Step-by-step explanation:

We are given that Z₁₅ is a group under addition modulo.

We have, H₁ = [0,5,10] and H₂ = [0,4,8,12].

First we check if H₁ is a subgroup of Z₁₅

We have,

• 0+0 = 0
• 0+5=5
• 0+10 = 10
• 5+0=5
• 5+5=10
• 5+10=0 (15=0 in Z₁₅)
• 10+0=10
• 10+5=0
• 10+10=5 (20=5 in Z₁₅)

Thus, we see that all the elements of H₁ after addition are also in H₁ and hence, H₁ is closed under addition. Moreover, we have, 0 is the identity element and 5+₁₅10 = 0, 10 + ₁₅5 = 0; and thus each element has its inverse element. Thus, H₁ is a subgroup of Z₁₅.

Now, we check if H₂ is a subgroup of Z₁₅

We have,

• 0+0=0
• 0+4=4
• 0+8=8
• 0+12=12
• 4+0=4
• 4+4=8
• 4+8=12
• 4+12=1 (16 = 1 in Z₁₅)
• 8+0=8
• 8+4=12
• 8+8=1
• 8+12=5
• 12+0=12
• 12+4=1
• 12+8=20
• 12+12=9 (24 = 9 in Z₁₅)

Here, in the last addition we see that the element 9 is not present in H₂.

Hence, H₂ is not closed under addition and thus H₂ is not a subgroup of Z₁₅.

Answer 2

Answer:

//nath27076 provided brief explanation

But the last addition of H2 , he has given 12+8=20 instead of 5 under the modulo addition.

Step-by-step explanation:

H₁ is a subgroup of Z₁₅ whereas H₂ is not a subgroup of Z₁₅

Step-by-step explanation:

We are given that Z₁₅ is a group under addition modulo.

We have, H₁ = [0,5,10] and H₂ = [0,4,8,12].

First we check if H₁ is a subgroup of Z₁₅

We have,

0+0 = 0

0+5=5

0+10 = 10

5+0=5

5+5=10

5+10=0 (15=0 in Z₁₅)

10+0=10

10+5=0

10+10=5 (20=5 in Z₁₅)

Thus, we see that all the elements of H₁ after addition are also in H₁ and hence, H₁ is closed under addition. Moreover, we have, 0 is the identity element and 5+₁₅10 = 0, 10 + ₁₅5 = 0; and thus each element has its inverse element. Thus, H₁ is a subgroup of Z₁₅.

Now, we check if H₂ is a subgroup of Z₁₅

We have,

0+0=0

0+4=4

0+8=8

0+12=12

4+0=4

4+4=8

4+8=12

4+12=1 (16 = 1 in Z₁₅)

8+0=8

8+4=12

8+8=1

8+12=5

12+0=12

12+4=1

12+8=5

12+12=9 (24 = 9 in Z₁₅)