Answer to Question #136600 in Abstract Algebra for emmanuel ehi

Question #136600
(a) Let M and T be a groups and φ : M −→ T an epimorphism. Prove that if M is Abelian then T is Abelian.
(b) Let G be a group and let H be a subgroup of index 2. Prove that for every g ∈ G, g−1 / ∈ H whenever g / ∈ H. Hence, prove that for all a,b ∈ G, if a / ∈ H and b / ∈ H then ab ∈ H.
(c) Find all the distinct left cosets of the subgroup H = h(1,1)i in Z2 ×Z4.
1
Expert's answer
2020-10-06T18:26:37-0400

(a) Let "a,b\\in T" be arbitrary. Since "\\varphi : M\\to T" is epimorphism, there are exist "x,y\\in M" such that "\\varphi(x)=a, \\varphi(y)=b." Taking into account that "xy=yx" for any "x,y\\in M" we conclude that

"ab=\\varphi(x)\\varphi(y)=\\varphi(xy)=\\varphi(yx)=\\varphi(y)\\varphi(x)=ba."

Therefore, T is Abelian.


(b) If "g\\notin H," then "g^{-1}\\notin H." Indeed, suppose that "g^{-1}\\in H". Since "H" is a group, "g=(g^{-1})^{-1}\\in H" and we have a contradiction.

Each subgroup "H" of index 2 is normal in "G" because "g_1H=G\\setminus H=Hg_2" for any "g_1\\notin H, g_2\\notin H." Consequenly, the quotient group "G\/H" exists and is of order 2. Thus, "(gH)(gH)=H" for any "g\\notin H." If "a\\notin H" and "b\\notin H", then "a,b\\in G\\setminus H=gH" for any "g\\notin H", and therefore, "ab\\in(gH)(gH)=H."


(с) Let "H=\\langle(1,1)\\rangle". Taking into account that "(1,1)+(1,1)=(0,2), (1,1)+(1,1)+(1,1)=(1,3)" and "(1,1)+(1,1)+(1,1)+(1,1)=(0,0)" we conclude that

"H=\\{(1,1), (0,2),(1,3),(0,0)\\}".


Let us find all the distinct left cosets:

"[(0,0)]=(0,0)+H=H,"

"[(0,1)]=(0,1)+H=\\{(1,2), (0,3),(1,0),(0,1)\\}."

Since "[(0,0)]\\cup[(0,1)]=\\mathbb Z_2\\times\\mathbb Z_4," there are 2 distinct left coset of the subgroup "H" in "\\mathbb Z_2\\times\\mathbb Z_4": "[(0,0)]" and "[(0,1)]."

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