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< G × H,*> is a group with identity (eg, eH).

< G × H,*> is a group with identity (eg, eH).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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3. Let < G,*1> and < H,*2> be two groups with identity elements eg and en· As you know the product of G and H is defined as: G x H = {(a,b)|a E G,b E H} Define an operation * on G × H as follows: (a, b) * (a', b') = (a *1 a', b *2 b') , where (a, b), (a', b') E G × H. Prove that < G × H,*> is a group with identity (eg, eH). 4. Prove that if G is abelian then H = {h E G|h¯1 = h} is a subgroup of the group G.
Transcribed Image Text:3. Let < G,*1> and < H,*2> be two groups with identity elements eg and en· As you know the product of G and H is defined as: G x H = {(a,b)|a E G,b E H} Define an operation * on G × H as follows: (a, b) * (a', b') = (a *1 a', b *2 b') , where (a, b), (a', b') E G × H. Prove that < G × H,*> is a group with identity (eg, eH). 4. Prove that if G is abelian then H = {h E G|h¯1 = h} is a subgroup of the group G.
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