- #1
nebbish
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Abstract Algebra -- group
Show that in a group G of order [tex]p^2[/tex] any normal subgroup of order p must lie in the center of G.
I am pretty sure here that p is supposed to be a prime number, as that is the stipulation in preceding and later problems. However, the problem statement does not explicitly state that p is prime here. I think that's just an oversight.
Attempted solution:
Let x be an element of N, a normal subgroup of G of order p. If x isn't e, then o(x)=p, as
o(x) must divide o(N). Then N = (x). Let [tex]x^r[/tex] be an arbitrary element of N. Let g be an arbitrary element of G.
Then by the normality of N, [tex]gx^rg^{-1}=x^s[/tex] for some s.
Then [tex]gx^r=x^sg[/tex]. If I could show r = s then I would be done.
Show that in a group G of order [tex]p^2[/tex] any normal subgroup of order p must lie in the center of G.
I am pretty sure here that p is supposed to be a prime number, as that is the stipulation in preceding and later problems. However, the problem statement does not explicitly state that p is prime here. I think that's just an oversight.
Attempted solution:
Let x be an element of N, a normal subgroup of G of order p. If x isn't e, then o(x)=p, as
o(x) must divide o(N). Then N = (x). Let [tex]x^r[/tex] be an arbitrary element of N. Let g be an arbitrary element of G.
Then by the normality of N, [tex]gx^rg^{-1}=x^s[/tex] for some s.
Then [tex]gx^r=x^sg[/tex]. If I could show r = s then I would be done.
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