Show that an intersection of normal subgroups of a group G is again a normal subgroup of G

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xandir307dc · Accepted Answer

Step 1 Solution: Given: Let H1 and H2 teo normal subgroup of G Prove: (H1∩H2) is normal subgroup of G Let H1∩H2≠ϕ Since at least the identity element &#039;e&#039; is the common to bot H1 and H2 In order to prove H1∩H2 is a subgroup, it is sufficient to prove that a∈H1∩H2 b∈H1∩H2 ⇒ab−1∈H1∩H2 Now a∈H1∩H2 and b∈H1∩H2 a∈H1 and ∈H2,b∈H1 and b∈H2 a∈H1 &amp;amp; b∈H1,a+H2, b∈H2 ⇒ab−1∈H1⇒ab−1∈H2 (∵H1 and H2 are normal subgroup) ab−1∈H1 &amp;amp; ab−1∈H2 ⇒ab−1∈H1∩H2 Hence, a∈H1∩H2,b∈H1∩H2 ⇒ab−1∈H1∩H2, here prove.

Steve Hirano · Accepted Answer

Step 1 Let G be a group and {Hi∣i∈I} be a family of normal subgroups of G. Let h∈∩IHi and g∈G. By the normality of each Hi we have that ghg−1∈Hi for all i∈I, so ghg−1∈∩IHi Therefore ∩IHi is normal.

alenahelenash · Accepted Answer

First show that H∩K is a subgroup of G using the fact that H and K are subgroups of G (forget for now that H,K are normal subgroups of G). For this you have to prove that ab−1∈H∩K for every a, b∈H∩K. Now you have to prove that H∩K is a normal subgroup. Here use the fact that H and K are normal subgroups of G. So you can show that if x∈G and y∈H∩K then xyx−1∈H∩K.

Show that an intersection of normal subgroups of a group

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