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mehtamonica
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I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.
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Thanks, Micromass. If G is the internal direct product of its subgroups H and K ,then the possible orders of subgroups H and K can be 2 and 4 or vice a versa.micromass said:So, what did you try already??
What can the orders be of a proper subgroup of D4?? Can they be abelian, nonabelian?
micromass said:Indeed, an the direct product of abelian groups is...
mehtamonica said:As far as the result goes the external direct product of two abelian groups is abelian...but is the internal direct product abelian too ? i mean if subgroups H and K are abelian can we conclude that the IDP is abelian ?
micromass said:Well, the internal direct product of H and G is isomorphic to the external direct product if [itex]H\cap G=\{e\}[/itex]. Use that.
A dihedral group is a mathematical concept that represents the symmetries of a regular polygon. It is denoted by Dn, where n represents the number of sides of the polygon. It is a group because it follows the four group axioms - closure, associativity, identity, and inverse elements.
The order of a dihedral group is equal to the number of elements in the group. It can be calculated using the formula |Dn| = 2n, where n is the number of sides of the polygon. For example, the order of a dihedral group with 4 sides (square) is |D4| = 2(4) = 8.
The internal direct product of a dihedral group is a way of combining two groups, Dm and Dn, to form a new group, Dm × Dn. It is defined as the set of all ordered pairs (g1, g2) where g1 is an element of Dm and g2 is an element of Dn, and the operation is defined as (g1, g2) × (h1, h2) = (g1h1, g2h2).
An example of an internal direct product of a dihedral group is D4 × D3. This is the group formed by combining a square (D4) and an equilateral triangle (D3). The order of this group is |D4 × D3| = (2)(4)(3) = 24, which is the product of the orders of the two individual groups.
The internal direct product of a dihedral group is a special case of the external direct product. While the internal direct product combines elements from the same group, the external direct product combines elements from different groups. The external direct product of two groups, G and H, is denoted by G × H and is defined as the set of all ordered pairs (g, h) where g is an element of G and h is an element of H, and the operation is defined as (g, h) × (g', h') = (gg', hh'). In contrast, the internal direct product of two groups, G and H, is denoted by G × H and is defined as the set of all ordered pairs (g, h) where g is an element of G and h is an element of H, and the operation is defined as (g, h) × (g', h') = (g×g', h×h').