Maximal subgroup of a product of groups?

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SUMMARY

The maximal subgroups of the product of two finite groups G and H, denoted as GxH, are characterized as GxM where M is a maximal subgroup of H, or NxG where N is a maximal subgroup of G. This assertion holds true in general, but fails in specific cases such as \(\mathbb{Z}_2 \times \mathbb{Z}_2\), where the subgroup {(0,0),(1,1)} is a maximal subgroup that does not conform to the stated forms. The discussion confirms the need for careful consideration of subgroup structures in group theory.

PREREQUISITES
  • Understanding of group theory and finite groups
  • Familiarity with maximal subgroups and their properties
  • Knowledge of direct products of groups
  • Basic concepts of algebraic structures in mathematics
NEXT STEPS
  • Study the properties of maximal subgroups in finite groups
  • Explore the structure of direct products of groups
  • Investigate specific examples of maximal subgroups in \(\mathbb{Z}_2 \times \mathbb{Z}_2\)
  • Learn about the classification of finite groups and their subgroup lattices
USEFUL FOR

Mathematicians, particularly those specializing in abstract algebra, group theorists, and students studying finite group structures will benefit from this discussion.

moont14263
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Let G and H be finite groups. The maximal subgroups of GxH are of the form GxM where M is maximal subgroup of H or NxG where N is a maximal subgroup of G. Is this true?
 
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Is it true in [itex]\mathbb{Z}_2\times \mathbb{Z}_2[/itex]??
 
not true as {(0,0),(1,1)} is a maximal subgroup of Z_2xZ_2 which is not of that form.
Thank you very much.
 

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